256
Chapter 3 | Derivatives
Simplifying, we obtain
2 +30 x
k ′( x ) = 20 x
.
(4 x +3) 2
Find the derivative of h ( x ) = 3 x +1 4 x −3 .
3.17
It is now possible to use the quotient rule to extend the power rule to find derivatives of functions of the form x k where k is a negative integer.
Theorem 3.7: Extended Power Rule If k is a negative integer, then
⎛ ⎝ x k
⎞ ⎠ = kx k −1 .
d dx
Proof If k is a negative integer, we may set n =− k , so that n is a positive integer with k =− n . Since for each positive integer n , x − n = 1 x n , we may now apply the quotient rule by setting f ( x ) =1 and g ( x ) = x n . In this case, f ′( x ) =0 and g ′( x ) = nx n −1 . Thus,
0( x n )−1 ⎛
⎞ ⎠
⎝ nx n −1
d dx
( x − n ) =
.
( x n ) 2
Simplifying, we see that
n −1
d dx
( x − n ) = − nx
=− nx ( n −1)−2 n =− nx − n −1 .
x 2 n
Finally, observe that since k =− n , by substituting we have d dx ⎛ ⎝ x k ⎞
⎠ = kx k −1 .
□
Example 3.26 Using the Extended Power Rule
⎛ ⎝ x −4
⎞ ⎠ .
Find d dx
Solution By applying the extended power rule with k =−4, we obtain
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