The first thing to notice when finding the derivative of this function is that it is a composition of two functions, as shown below:
| V | = | cos( |
| where | |
= | 9*t - 4 |
The Chain Rule (Derivative Rule for Compositions):
If
|
Or, if
|
|||||||||||||||||||||||||||||||||
So our example,
| V | = | cos( |
| where | |
= | 9*t - 4 |
| V | = | f( |
, where | |
= | g(t) | = | 9*t - 4 |
| V ' | = ( | f( |
)' | |
| = | f '( |
( |
||
| = | ( cos( |
( 9*t - 4 )' | ||
| ( cos( |
= | (-1)*sin( |
(by the derivative rules for basic functions) |
| ( 9*t - 4 )' | = | ( 9 - 0 ) | (by the derivative rule for sums, rule for constant multiples, and the derivative rule for constants) |
| V ' | = | (-1)*sin( |
( 9 - 0 ) |
| = | (-1)*sin(9*t - 4) | ( 9 - 0 ) | |
| = | (-9)*sin(9*t - 4) | ||