Chapter 3 | Derivatives
257
⎛ ⎝ x −4
⎞ ⎠ =−4 x −4−1 =−4 x −5 .
d dx
Example 3.27 Using the Extended Power Rule and the Constant Multiple Rule
Use the extended power rule and the constant multiple rule to find the derivative of f ( x ) = 6 x 2 .
Solution It may seem tempting to use the quotient rule to find this derivative, and it would certainly not be incorrect to do so. However, it is far easier to differentiate this function by first rewriting it as f ( x ) =6 x −2 . f ′( x ) = d dx ⎛ ⎝ 6 x 2 ⎞ ⎠ = d dx ⎛ ⎝ 6 x −2 ⎞ ⎠ Rewrite 6 x 2 as6 x −2 .
=6 d
x −2 )
dx (
Apply the constant multiple rule.
=6(−2 x −3 ) =−12 x −3
Use the extended power rule to differentiate x −2 .
Simplify.
3.18
Find the derivative of g ( x ) = 1 x 7
using the extended power rule.
Combining Differentiation Rules As we have seen throughout the examples in this section, it seldom happens that we are called on to apply just one differentiation rule to find the derivative of a given function. At this point, by combining the differentiation rules, we may find the derivatives of any polynomial or rational function. Later on we will encounter more complex combinations of differentiation rules. A good rule of thumb to use when applying several rules is to apply the rules in reverse of the order in which we would evaluate the function. Example 3.28 Combining Differentiation Rules
For k ( x ) =3 h ( x )+ x 2 g ( x ), find k ′( x ).
Solution Finding this derivative requires the sum rule, the constant multiple rule, and the product rule.
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