# Limit Definition of the Derivative by mikeholy

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```									                          Harvey Mudd College Math Tutorial:

Limit Deﬁnition of the Derivative
Once we know the most basic diﬀerentiation formulas and rules, we compute new derivatives
using what we already know. We rarely think back to where the basic formulas and rules
originated.

The geometric meaning of the derivative

df (x)
f (x) =
dx
is the slope of the line tangent to y = f (x) at x.

Let’s look for this slope at P :

The secant line through P and Q has
slope

f (x + ∆x) − f (x)   f (x + ∆x) − f (x)
=                    .
(x + ∆x) − x             ∆x

We can approximate the tangent line through P by moving Q towards P , decreasing ∆x.
In the limit as ∆x → 0, we get the tangent line through P with slope

f (x + ∆x) − f (x)
lim                       .
∆x→0           ∆x
We deﬁne
f (x + ∆x) − f (x) ∗
f (x) = lim                      .
∆x→0        ∆x

∗
If the limit as ∆x → 0 at a particular point does not exist, f (x) is undeﬁned at that point.

We derive all the basic diﬀerentiation formulas using this deﬁnition.

Example

For f (x) = x2 ,

(x + ∆x)2 − x2
f (x) =     lim
∆x→0      ∆x
(x2 + 2(∆x)x + ∆x2 ) − x2
=    lim
∆x→0           ∆x
2(∆x)x + ∆x2
= lim
∆x→0      ∆x
= lim (2x + ∆x)
∆x→0
= 2x

as expected.

Example

1
For f (x) =
x
1     1
x+∆x
−   x
f (x) =     lim
∆x→0      ∆x
x−(x+∆x)
(x+∆x)(x)
=    lim
∆x→0     ∆x
−∆x
(x+∆x)(x)
=    lim
∆x→0  ∆x
−1
= lim
∆x→0 (x + ∆x)(x)
1
= − 2
x
again as expected.

Notes

The limit deﬁnition of the derivative is used to prove many well-known results, including the
following:

• If f is diﬀerentiable at x0 , then f is continuous at x0 .
d n
• Diﬀerentiation of polynomials:       [x ] = nxn−1 .
dx
• Product and Quotient Rules for diﬀerentiation.
Key Concepts

f (x + ∆x) − f (x)
We deﬁne f (x) = lim                      .
∆x→0          ∆x
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