The first thing to notice when finding the derivative of this function is that it is a composition of two functions, as shown below:
| C(x) | = | ln( |
| where | |
= | 3*x5 |
The Chain Rule (Derivative Rule for Compositions):
If
|
Or, if
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So our example,
| C(x) | = | ln( |
| where | |
= | 3*x5 |
| C(x) | = | f( |
, where | |
= | g(x) | = | 3*x5 |
| C '(x) | = ( | f( |
)' | |
| = | f '( |
( |
||
| = | ( ln( |
( 3*x5 )' | ||
| ( ln( |
= | ( |
(by the derivative rules for basic functions) |
| ( 3*x5 )' | = | ( 3*5*x4 ) | (by the rule for constant multiples, and the power rule) |
| C '(x) | = | ( |
( 3*5*x4 ) |
| = | (3*x5)-1 | ( 3*5*x4 ) | |
| = | 15*x4 (3*x5)-1 | ||