The first thing to notice when finding the derivative of this function is that it is a composition of two functions, as shown below:
| y | = | ( |
| where | |
= | sin(t) |
The Chain Rule (Derivative Rule for Compositions):
If
|
Or, if
|
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So our example,
| y | = | ( |
| where | |
= | sin(t) |
| y | = | f( |
, where | |
= | g(t) | = | sin(t) |
| y ' | = ( | f( |
)' | |
| = | f '( |
( |
||
| = | ( ( |
( sin(t) )' | ||
| ( ( |
= | (1/4)*( |
(by the power rule) |
| ( sin(t) )' | = | ( cos(t) ) | (by the derivative rules for basic functions) |
| y ' | = | (1/4)*( |
( cos(t) ) |
| = | (1/4)*(sin(t))-3/4 | ( cos(t) ) | |
| = | (1/4)*(sin(t))-3/4 cos(t) | ||