250
Chapter 3 | Derivatives
⎛ ⎝
⎞ ⎠ x n −3 h 3 +…+ nxh n −1 + h n .
( x + h ) n − x n = nx n −1 h + ⎛ ⎝
⎞ ⎠ x n −2 h 2 +
n 3
n 2
Next, divide both sides by h :
⎛ ⎝
⎞ ⎠ x n −3 h 3 +…+ nxh n −1 + h n h .
⎛ ⎝
⎞ ⎠ x n −2 h 2 +
n 3
n 2
nx n −1 h +
( x + h ) n − x n h
=
Thus,
⎛ ⎝
⎞ ⎠ x n −3 h 2 +…+ nxh n −2 + h n −1 .
⎛ ⎝
⎞ ⎠ x n −2 h +
( x + h ) n − x n h
n 3
n 2
nx n −1 +
=
Finally,
f ′( x ) = lim h →0 ⎛
⎛ ⎝
⎞ ⎠ x n −3 h 2 +…+ nxh n −1 + h n ⎞ ⎠
⎛ ⎝
⎞ ⎠ x n −2 h +
n 3
n 2
⎝ nx n −1 +
= nx n −1 .
□
Example 3.19 Applying the Power Rule
Find the derivative of the function f ( x ) = x 10 by applying the power rule.
Solution Using the power rule with n =10, we obtain
f ′( x ) =10 x 10−1 =10 x 9 .
Find the derivative of f ( x ) = x 7 .
3.13
The Sum, Difference, and Constant Multiple Rules We find our next differentiation rules by looking at derivatives of sums, differences, and constant multiples of functions. Just as when we work with functions, there are rules that make it easier to find derivatives of functions that we add, subtract, or multiply by a constant. These rules are summarized in the following theorem. Theorem 3.4: Sum, Difference, and Constant Multiple Rules Let f ( x ) and g ( x ) be differentiable functions and k be a constant. Then each of the following equations holds. SumRule . The derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g . d dx ⎛ ⎝ f ( x )+ g ( x ) ⎞ ⎠ = d dx ⎛ ⎝ f ( x ) ⎞ ⎠ + d dx ⎛ ⎝ g ( x ) ⎞ ⎠ ; that is, for j ( x ) = f ( x )+ g ( x ), j ′( x ) = f ′( x )+ g ′( x ). Difference Rule . The derivative of the difference of a function f and a function g is the same as the difference of the
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