Chapter 3 | Derivatives
233
Substitute f ( x + h ) = x + h and f ( x ) = x into f ′( x ) = lim h →0 f ( x + h )− f ( x ) h . Multiply numerator and denominator by x + h + x without distributing in the denominator. Multiply the numerators and simplify.
x + h − x h
f ′( x ) = lim h →0
x + h − x h
x + h + x x + h + x
= lim
·
h →0
h h ( x + h + x ) 1 ( x + h + x )
= lim
h →0
= lim
Cancel the h .
h →0
= 1 2 x
Evaluate the limit.
Example 3.12 Finding the Derivative of a Quadratic Function
Find the derivative of the function f ( x ) = x 2 −2 x .
Solution Follow the same procedure here, but without having to multiply by the conjugate.
Substitute f ( x + h ) = ( x + h ) 2 −2( x + h ) and f ( x ) = x 2 −2 x into f ′( x ) = lim h →0 f ( x + h )− f ( x ) h .
(( x + h ) 2 −2( x + h ))−( x 2 −2 x ) h
f ′( x ) = lim h →0
x 2 +2 xh + h 2 −2 x −2 h − x 2 +2 x h
x + h ) 2 −2( x + h ).
= lim
Expand(
h →0
2 xh −2 h + h 2 h h (2 x −2+ h ) h (2 x −2+ h )
= lim
Simplify.
h →0
= lim = lim
Factor out
h from the numerator.
h →0
Cancel the common factor of h .
h →0
=2 x −2
Evaluate the limit.
Find the derivative of f ( x ) = x 2 .
3.6
We use a variety of different notations to express the derivative of a function. In Example 3.12 we showed that if f ( x ) = x 2 −2 x , then f ′( x ) =2 x −2. If we had expressed this function in the form y = x 2 −2 x , we could have expressed the derivative as y ′ =2 x −2 or dy dx =2 x −2. We could have conveyed the same information by writing d dx ⎛ ⎝ x 2 −2 x ⎞ ⎠ =2 x −2. Thus, for the function y = f ( x ), each of the following notations represents the derivative of f ( x ):
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