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 | |
= | 10*t2 + p |
The Chain Rule (Derivative Rule for Compositions):
If
|
Or, if
|
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So our example,
| y | = | ( |
| where | |
= | 10*t2 + p |
| y | = | f( |
, where | |
= | g(t) | = | 10*t2 + p |
| y ' | = ( | f( |
)' | |
| = | f '( |
( |
||
| = | ( ( |
( 10*t2 + p )' | ||
| ( ( |
= | (p)*( |
(by the power rule) |
| ( 10*t2 + p )' | = | ( 10*2*t + 0 ) | (by the derivative rule for sums, rule for constant multiples, and the derivative rule for constants) |
| y ' | = | (p)*( |
( 10*2*t + 0 ) |
| = | (p)*(10*t2 + p)p-1 | ( 10*2*t + 0 ) | |
| = | (20p)*t (10*t2 + p)p-1 | ||