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Introduction Derivation The Rules Examples
The Product and Quotient Rules
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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
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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
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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
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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
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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
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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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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)
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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
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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
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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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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
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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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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
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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 + b
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 · 1 + b
b = −e
y = 2ex − 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
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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
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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
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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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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
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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
Find Where f (x) = is Increasing or
x2 + 4
Decreasing
f (x)
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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
Find Where f (x) = is Increasing or
x2 + 4
Decreasing
1 · x2 + 4 − 2x · x
f (x) = 2
(x2 + 4)
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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
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)
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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
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
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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
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
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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
f (x)
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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
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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
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
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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
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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Bernd Schr¨ der Louisiana Tech University, College of Engineering and Science
The Product and Quotient Rules
