# The Product and Quotient Rules

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```							Introduction                     Derivation                   The Rules                               Examples

The Product and Quotient Rules

o
Bernd Schr¨ der

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Bernd Schr¨ der                                    Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Why Do we Need the Product Rule and the
Quotient Rule?

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Why Do we Need the Product Rule and the
Quotient Rule?
Products like ex x2 + 2 cannot be multiplied out like, say,
x3 x2 + 2 .

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Why Do we Need the Product Rule and the
Quotient Rule?
Products like ex x2 + 2 cannot be multiplied out like, say,
x3 x2 + 2 .

2x2 + 3
Similarly, quotients like        cannot be converted into
x2 − 4
functions that do not involve quotients.

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

What Should the Product Rule and the Quotient
Rule Look Like?

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful.

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients

logo1
o
Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3

logo1
o
Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                     The Rules                               Examples

What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3           k(x) := x2 + 2

logo1
o
Bernd Schr¨ der                                      Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                       Derivation                     The Rules                               Examples

What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3             k(x) := x2 + 2
gk(x) = x3 x2 + 2

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o
Bernd Schr¨ der                                        Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                       Derivation                     The Rules                               Examples

What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3             k(x) := x2 + 2
gk(x) = x3 x2 + 2 = x5 + 2x3

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Bernd Schr¨ der                                        Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                       Derivation                     The Rules                               Examples

What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3             k(x) := x2 + 2
gk(x) = x3 x2 + 2 = x5 + 2x3
(gk) (x) = 5x4 + 6x2

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o
Bernd Schr¨ der                                        Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                       Derivation                     The Rules                               Examples

What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3             k(x) := x2 + 2
gk(x) = x3 x2 + 2 = x5 + 2x3
(gk) (x) = 5x4 + 6x2
= g (x)k (x)

logo1
o
Bernd Schr¨ der                                        Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                       Derivation                     The Rules                               Examples

What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3             k(x) := x2 + 2
gk(x) = x3 x2 + 2 = x5 + 2x3
(gk) (x) = 5x4 + 6x2
= g (x)k (x) = 3x2

logo1
o
Bernd Schr¨ der                                        Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                       Derivation                     The Rules                               Examples

What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3             k(x) := x2 + 2
gk(x) = x3 x2 + 2 = x5 + 2x3
(gk) (x) = 5x4 + 6x2
= g (x)k (x) = 3x2 · 2x

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Bernd Schr¨ der                                        Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                       Derivation                     The Rules                               Examples

What Should the Product Rule and the Quotient
Rule Look Like?
We must be careful. Derivatives cannot simply be moved into
products and quotients:
g(x) := x3             k(x) := x2 + 2
gk(x) = x3 x2 + 2 = x5 + 2x3
(gk) (x) = 5x4 + 6x2
= g (x)k (x) = 3x2 · 2x = 6x3

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Bernd Schr¨ der                                        Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Deriving the Product Rule.

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
(gk) (x) = lim
h→0           h

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Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                The Rules                               Examples

Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
(gk) (x) = lim
h→0           h
g(x + h)k(x + h) − g(x)k(x)
= lim
h→0              h

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o
Bernd Schr¨ der                                 Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                 The Rules                               Examples

Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
(gk) (x) = lim
h→0           h
g(x + h)k(x + h) − g(x)k(x)
= lim
h→0              h
g(x + h)k(x + h) − g(x)k(x + h) + g(x)k(x + h) − g(x)k(x)
= lim
h→0                             h

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Bernd Schr¨ der                                  Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                 The Rules                               Examples

Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
(gk) (x) = lim
h→0           h
g(x + h)k(x + h) − g(x)k(x)
= lim
h→0              h
g(x + h)k(x + h) − g(x)k(x + h) + g(x)k(x + h) − g(x)k(x)
= lim
h→0                             h
g(x + h) − g(x) k(x + h) + k(x + h) − k(x) g(x)
= lim
h→0                         h

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Bernd Schr¨ der                                  Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
(gk) (x) = lim
h→0            h
g(x + h)k(x + h) − g(x)k(x)
=   lim
h→0              h
g(x + h)k(x + h) − g(x)k(x + h) + g(x)k(x + h) − g(x)k(x)
=   lim
h→0                              h
g(x + h) − g(x) k(x + h) + k(x + h) − k(x) g(x)
=   lim
h→0                          h
g(x + h) − g(x)            k(x + h) − k(x)
=   lim                   k(x + h) +                 g(x)
h→0          h                          h

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Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
(gk) (x) = lim
h→0            h
g(x + h)k(x + h) − g(x)k(x)
=   lim
h→0              h
g(x + h)k(x + h) − g(x)k(x + h) + g(x)k(x + h) − g(x)k(x)
=   lim
h→0                              h
g(x + h) − g(x) k(x + h) + k(x + h) − k(x) g(x)
=   lim
h→0                          h
g(x + h) − g(x)            k(x + h) − k(x)
=   lim                   k(x + h) +                 g(x)
h→0          h                          h
g(x + h) − g(x)                    k(x + h) − k(x)
=   lim                 lim k(x + h) + lim                 lim g(x)
h→0        h        h→0            h→0        h        h→0

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Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
(gk) (x) = lim
h→0            h
g(x + h)k(x + h) − g(x)k(x)
=   lim
h→0               h
g(x + h)k(x + h) − g(x)k(x + h) + g(x)k(x + h) − g(x)k(x)
=   lim
h→0                              h
g(x + h) − g(x) k(x + h) + k(x + h) − k(x) g(x)
=   lim
h→0                          h
g(x + h) − g(x)            k(x + h) − k(x)
=   lim                   k(x + h) +                 g(x)
h→0          h                          h
g(x + h) − g(x)                    k(x + h) − k(x)
=   lim                 lim k(x + h) + lim                 lim g(x)
h→0        h        h→0            h→0        h        h→0
=   g (x)k(x) + k (x)g(x)

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Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Deriving the Product Rule.
(gk)(x + h) − (gk)(x)
(gk) (x) = lim
h→0            h
g(x + h)k(x + h) − g(x)k(x)
=   lim
h→0               h
g(x + h)k(x + h) − g(x)k(x + h) + g(x)k(x + h) − g(x)k(x)
=   lim
h→0                              h
g(x + h) − g(x) k(x + h) + k(x + h) − k(x) g(x)
=   lim
h→0                          h
g(x + h) − g(x)            k(x + h) − k(x)
=   lim                   k(x + h) +                 g(x)
h→0          h                          h
g(x + h) − g(x)                    k(x + h) − k(x)
=   lim                 lim k(x + h) + lim                 lim g(x)
h→0        h        h→0            h→0        h        h→0
=   g (x)k(x) + k (x)g(x)
The derivation of the quotient rule is similar.
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Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Theorem.

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Theorem. The product rule and the quotient rule.

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Theorem. The product rule and the quotient rule. If p = gk
g
and q = and g and k are differentiable at the point x
k

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Theorem. The product rule and the quotient rule. If p = gk
g
and q = and g and k are differentiable at the point x, then
k
p (x) = (gk) (x) = g (x)k(x) + k (x)g(x),

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                      The Rules                               Examples

Theorem. The product rule and the quotient rule. If p = gk
g
and q = and g and k are differentiable at the point x, then
k
p (x) = (gk) (x) = g (x)k(x) + k (x)g(x),
and if k(x) = 0, then
g             g (x)k(x) − k (x)g(x)
q (x) =            (x) =                         .
k                     k2 (x)

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Bernd Schr¨ der                                          Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                      The Rules                               Examples

Theorem. The product rule and the quotient rule. If p = gk
g
and q = and g and k are differentiable at the point x, then
k
p (x) = (gk) (x) = g (x)k(x) + k (x)g(x),
and if k(x) = 0, then
g             g (x)k(x) − k (x)g(x)
q (x) =            (x) =                         .
k                     k2 (x)
Remember the product rule in the same order as the numerator
of the quotient rule.

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Bernd Schr¨ der                                          Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                      The Rules                               Examples

Theorem. The product rule and the quotient rule. If p = gk
g
and q = and g and k are differentiable at the point x, then
k
p (x) = (gk) (x) = g (x)k(x) + k (x)g(x),
and if k(x) = 0, then
g             g (x)k(x) − k (x)g(x)
q (x) =            (x) =                         .
k                     k2 (x)
Remember the product rule in the same order as the numerator
of the quotient rule.
Product The derivative of a product is the derivative of the ﬁrst factor times
Rule.   the second factor plus the derivative of the second factor times the
ﬁrst factor.

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Bernd Schr¨ der                                          Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                      The Rules                               Examples

Theorem. The product rule and the quotient rule. If p = gk
g
and q = and g and k are differentiable at the point x, then
k
p (x) = (gk) (x) = g (x)k(x) + k (x)g(x),
and if k(x) = 0, then
g             g (x)k(x) − k (x)g(x)
q (x) =            (x) =                         .
k                     k2 (x)
Remember the product rule in the same order as the numerator
of the quotient rule.
Product The derivative of a product is the derivative of the ﬁrst factor times
Rule.    the second factor plus the derivative of the second factor times the
ﬁrst factor.
Quotient The derivative of a quotient is the derivative of the numerator times
Rule.    the denominator minus the derivative of the denominator times the
numerator (as a quantity) divided by the square of the denominator.
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Bernd Schr¨ der                                          Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Compute the derivative of f (x) = ex x2 + 2

logo1
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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation              The Rules                               Examples

Compute the derivative of f (x) = ex x2 + 2

f (x)

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Bernd Schr¨ der                                  Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation              The Rules                               Examples

Compute the derivative of f (x) = ex x2 + 2

f (x) = (ex )

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Bernd Schr¨ der                                Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                   The Rules                               Examples

Compute the derivative of f (x) = ex x2 + 2

f (x) = (ex )      x2 + 2

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Bernd Schr¨ der                                     Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                  The Rules                               Examples

Compute the derivative of f (x) = ex x2 + 2

f (x) = (ex )      x2 + 2 + x2 + 2

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Bernd Schr¨ der                                    Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                  The Rules                               Examples

Compute the derivative of f (x) = ex x2 + 2

f (x) = (ex )      x2 + 2 + x2 + 2 ex

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Bernd Schr¨ der                                    Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                  The Rules                               Examples

Compute the derivative of f (x) = ex x2 + 2

f (x) = (ex )      x2 + 2 + x2 + 2 ex

= ex

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Bernd Schr¨ der                                    Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                  The Rules                               Examples

Compute the derivative of f (x) = ex x2 + 2

f (x) = (ex )      x2 + 2 + x2 + 2 ex

= ex x2 + 2

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Bernd Schr¨ der                                    Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                  The Rules                               Examples

Compute the derivative of f (x) = ex x2 + 2

f (x) = (ex )      x2 + 2 + x2 + 2 ex

= ex x2 + 2 + 2x

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Bernd Schr¨ der                                    Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                  The Rules                               Examples

Compute the derivative of f (x) = ex x2 + 2

f (x) = (ex )      x2 + 2 + x2 + 2 ex

= ex x2 + 2 + 2xex

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Bernd Schr¨ der                                    Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                  The Rules                               Examples

Compute the derivative of f (x) = ex x2 + 2

f (x) = (ex )      x2 + 2 + x2 + 2 ex

= ex x2 + 2 + 2xex

= ex x2 + 2x + 2

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Bernd Schr¨ der                                    Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

2x2 + 3
Compute the derivative of f (x) = 2
x −4

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

2x2 + 3
Compute the derivative of f (x) = 2
x −4

f (x)

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                   The Rules                               Examples

2x2 + 3
Compute the derivative of f (x) = 2
x −4
2x2 + 3      x2 − 4 − x2 − 4                2x2 + 3
f (x) =                                     2
(x2 − 4)

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Bernd Schr¨ der                                    Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                         The Rules                               Examples

2x2 + 3
Compute the derivative of f (x) = 2
x −4
2x2 + 3         x2 − 4 − x2 − 4                   2x2 + 3
f (x) =                                            2
(x2 − 4)
4x x2 − 4 − 2x 2x2 + 3
=                        2
(x2 − 4)

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Bernd Schr¨ der                                           Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                       The Rules                               Examples

2x2 + 3
Compute the derivative of f (x) = 2
x −4
2x2 + 3      x2 − 4 − x2 − 4                    2x2 + 3
f (x) =                                          2
(x2 − 4)
4x x2 − 4 − 2x 2x2 + 3
=                      2
(x2 − 4)
4x3 − 16x − 4x3 − 6x
=                  2
(x2 − 4)

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Bernd Schr¨ der                                         Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                           The Rules                               Examples

2x2 + 3
Compute the derivative of f (x) = 2
x −4
2x2 + 3          x2 − 4 − x2 − 4                    2x2 + 3
f (x) =                                              2
(x2 − 4)
4x x2 − 4 − 2x 2x2 + 3
=                          2
(x2 − 4)
4x3 − 16x − 4x3 − 6x
=                      2
(x2 − 4)
−22x
=                 2
(x2 − 4)

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Bernd Schr¨ der                                             Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                           The Rules                               Examples

2x2 + 3
Compute the derivative of f (x) = 2
x −4
2x2 + 3          x2 − 4 − x2 − 4                    2x2 + 3
f (x) =                                              2
(x2 − 4)
4x x2 − 4 − 2x 2x2 + 3
=                          2
(x2 − 4)
4x3 − 16x − 4x3 − 6x
=                      2
(x2 − 4)
−22x
=                 2
(x2 − 4)
(Leave the denominator as a power.)
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Bernd Schr¨ der                                             Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x)

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Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                 The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) = 1 · ex

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Bernd Schr¨ der                                  Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                 The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) = 1 · ex + xex

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Bernd Schr¨ der                                  Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                 The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) = 1 · ex + xex
= ex (x + 1)

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Bernd Schr¨ der                                  Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) = 1 · ex + xex
= ex (x + 1)
f (1)

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Bernd Schr¨ der                                 Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) = 1 · ex + xex
= ex (x + 1)
f (1) = 2e

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Bernd Schr¨ der                                 Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) = 1 · ex + xex
= ex (x + 1)
f (1) = 2e
f (1)

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Bernd Schr¨ der                                 Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) =    1 · ex + xex
=   ex (x + 1)
f (1) =    2e
f (1) =   e

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Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) =    1 · ex + xex
=   ex (x + 1)
f (1) =    2e
f (1) =   e
y =   mx + b

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Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) =    1 · ex + xex
=   ex (x + 1)
f (1) =    2e
f (1) =   e
y =   mx + b
e

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Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) =    1 · ex + xex
=   ex (x + 1)
f (1) =    2e
f (1) =   e
y =   mx + b
e =   2e

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Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) =    1 · ex + xex
=   ex (x + 1)
f (1) =    2e
f (1) =   e
y =   mx + b
e =   2e · 1

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o
Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) =    1 · ex + xex
=   ex (x + 1)
f (1) =    2e
f (1) =   e
y =   mx + b
e =   2e · 1 + b

logo1
o
Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) =    1 · ex + xex
=   ex (x + 1)
f (1) =    2e
f (1) =   e
y =   mx + b
e =   2e · 1 + b
b =   −e

logo1
o
Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                  The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

f (x) =    1 · ex + xex
=   ex (x + 1)
f (1) =    2e
f (1) =   e
y =   mx + b
e =   2e · 1 + b
b =   −e
y =   2ex − e

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o
Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

logo1
o
Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

logo1
o
Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

logo1
o
Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

Find the Equation of the Tangent Line of
f (x) = xex at a = 1

logo1
o
Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                 The Rules                               Examples

x
Find Where f (x) =                               is Increasing or
x2 + 4
Decreasing

logo1
o
Bernd Schr¨ der                                  Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                 The Rules                               Examples

x
Find Where f (x) =                                is Increasing or
x2 + 4
Decreasing

f (x)

logo1
o
Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                         The Rules                               Examples

x
Find Where f (x) =                               is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                    2
(x2 + 4)

logo1
o
Bernd Schr¨ der                                          Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                      The Rules                               Examples

x
Find Where f (x) =                               is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                  2
(x2 + 4)
4 − x2
=            2
(x2 + 4)

logo1
o
Bernd Schr¨ der                                       Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                   The Rules                               Examples

x
Find Where f (x) =                                is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                   2
(x2 + 4)
4 − x2       (2 − x)(2 + x)
=            =
(x 2 + 4)2       (x2 + 4)
2

logo1
o
Bernd Schr¨ der                                     Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                   The Rules                               Examples

x
Find Where f (x) =                                is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                   2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

logo1
o
Bernd Schr¨ der                                     Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                   The Rules                               Examples

x
Find Where f (x) =                                  is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                     2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

logo1
o
Bernd Schr¨ der                                       Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                   The Rules                               Examples

x
Find Where f (x) =                                  is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                     2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

-

logo1
o
Bernd Schr¨ der                                       Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                   The Rules                               Examples

x
Find Where f (x) =                                  is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                     2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

-

logo1
o
Bernd Schr¨ der                                       Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                   The Rules                               Examples

x
Find Where f (x) =                                  is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                     2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

-

logo1
o
Bernd Schr¨ der                                       Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                   The Rules                               Examples

x
Find Where f (x) =                                  is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                     2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

-
−2

logo1
o
Bernd Schr¨ der                                       Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                   The Rules                               Examples

x
Find Where f (x) =                                  is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                     2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

-
−2                          2

logo1
o
Bernd Schr¨ der                                       Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                   The Rules                               Examples

x
Find Where f (x) =                                  is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                     2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

-
−2                          2

logo1
o
Bernd Schr¨ der                                       Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                   The Rules                               Examples

x
Find Where f (x) =                                  is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                     2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

-
−2                          2

logo1
o
Bernd Schr¨ der                                       Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                   The Rules                               Examples

x
Find Where f (x) =                                  is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                     2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

-
−2                          2

logo1
o
Bernd Schr¨ der                                       Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                        Derivation                   The Rules                               Examples

x
Find Where f (x) =                                  is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                     2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

-
−3    −2                          2

logo1
o
Bernd Schr¨ der                                       Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                           Derivation                   The Rules                               Examples

x
Find Where f (x) =                                     is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                        2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

-
−3    −2            0                2

logo1
o
Bernd Schr¨ der                                          Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                           Derivation                      The Rules                               Examples

x
Find Where f (x) =                                     is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                           2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

-
−3    −2            0                2   3

logo1
o
Bernd Schr¨ der                                             Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                           Derivation                      The Rules                               Examples

x
Find Where f (x) =                                     is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                           2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

f
-
−3    −2            0                2   3

logo1
o
Bernd Schr¨ der                                             Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                           Derivation                      The Rules                               Examples

x
Find Where f (x) =                                     is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                           2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2   = ±2

f       −−−
-
−3    −2            0                2   3

logo1
o
Bernd Schr¨ der                                             Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                            Derivation                      The Rules                               Examples

x
Find Where f (x) =                                      is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                            2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2    = ±2

f       −−−                 +++
-
−3    −2             0                2   3

logo1
o
Bernd Schr¨ der                                              Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                            Derivation                        The Rules                               Examples

x
Find Where f (x) =                                      is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                              2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2    = ±2

f       −−−                 +++                   −−−
-
−3    −2             0                2    3

logo1
o
Bernd Schr¨ der                                                Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                            Derivation                        The Rules                               Examples

x
Find Where f (x) =                                      is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                              2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2    = ±2
f
f       −−−                 +++                   −−−
-
−3    −2             0                2    3

logo1
o
Bernd Schr¨ der                                                Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                            Derivation                        The Rules                               Examples

x
Find Where f (x) =                                      is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                            2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2    = ±2
f decreasing
f    −−−                    +++                   −−−
-
−3      −2           0                2    3

logo1
o
Bernd Schr¨ der                                                Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                             Derivation                        The Rules                               Examples

x
Find Where f (x) =                                       is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                             2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2     = ±2
f decreasing             increasing
f    −−−                    +++                    −−−
-
−3      −2            0                2    3

logo1
o
Bernd Schr¨ der                                                 Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                             Derivation                               The Rules                               Examples

x
Find Where f (x) =                                       is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) =                                    2
(x2 + 4)
4 − x2       (2 − x)(2 + x) !
=            =                 =0
(x 2 + 4)2       (x2 + 4)
2

x1,2     = ±2
f decreasing             increasing               decreasing
f    −−−                    +++                      −−−
-
−3      −2            0                2       3

logo1
o
Bernd Schr¨ der                                                        Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                 The Rules                               Examples

x
Find Where f (x) =                               is Increasing or
x2 + 4
Decreasing

logo1
o
Bernd Schr¨ der                                  Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                 The Rules                               Examples

x
Find Where f (x) =                               is Increasing or
x2 + 4
Decreasing

logo1
o
Bernd Schr¨ der                                  Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                             Derivation                           The Rules                               Examples

x
Find Where f (x) =                                       is Increasing or
x2 + 4
Decreasing

f      decreasing        increasing           decreasing
f         −−−               +++                  −−−
-
−3        −2       0            2       3
logo1
o
Bernd Schr¨ der                                                    Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                             Derivation                           The Rules                               Examples

x
Find Where f (x) =                                       is Increasing or
x2 + 4
Decreasing

f      decreasing        increasing           decreasing
f         −−−               +++                  −−−
-
−3        −2       0            2       3
logo1
o
Bernd Schr¨ der                                                    Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                             Derivation                           The Rules                               Examples

x
Find Where f (x) =                                       is Increasing or
x2 + 4
Decreasing

f      decreasing        increasing           decreasing
f         −−−               +++                  −−−
-
−3        −2       0            2       3
logo1
o
Bernd Schr¨ der                                                    Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation              The Rules                               Examples

x 2 ex
Compute the Derivative of f (x) =
x+1

logo1
o
Bernd Schr¨ der                               Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation              The Rules                               Examples

x 2 ex
Compute the Derivative of f (x) =
x+1

f (x)

logo1
o
Bernd Schr¨ der                                Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                     Derivation                    The Rules                               Examples

x 2 ex
Compute the Derivative of f (x) =
x+1
2xex + ex x2 (x + 1) − 1 · x2 ex
f (x) =
(x + 1)2

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o
Bernd Schr¨ der                                     Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                 The Rules                               Examples

x 2 ex
Compute the Derivative of f (x) =
x+1
2xex + ex x2 (x + 1) − 1 · x2 ex
f (x) =
(x + 1)2
ex 2x2 + x3 + 2x + x2 − x2 ex
=
(x + 1)2

logo1
o
Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
Introduction                      Derivation                 The Rules                               Examples

x 2 ex
Compute the Derivative of f (x) =
x+1
2xex + ex x2 (x + 1) − 1 · x2 ex
f (x) =
(x + 1)2
ex 2x2 + x3 + 2x + x2 − x2 ex
=
(x + 1)2
ex x3 + 2x2 + 2x
=
(x + 1)2

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o
Bernd Schr¨ der                                   Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules

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