The first thing to notice when finding the derivative of this function is that it is a composition of two functions, as shown below:
| f(x) | = | ln( |
| where | |
= | ex + x-1 |
The Chain Rule (Derivative Rule for Compositions):
If
|
Or, if
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So our example,
| f(x) | = | ln( |
| where | |
= | ex + x-1 |
| f(x) | = | f( |
, where | |
= | g(x) | = | ex + x-1 |
| f '(x) | = ( | f( |
)' | |
| = | f '( |
( |
||
| = | ( ln( |
( ex + x-1 )' | ||
| ( ln( |
= | ( |
(by the derivative rules for basic functions) |
| ( ex + x-1 )' | = | ( ex + (-1)*x-2 ) | (by the derivative rule for sums, derivative rules for basic functions, and the power rule) |
| f '(x) | = | ( |
( ex + (-1)*x-2 ) |
| = | (ex + x-1)-1 | ( ex + (-1)*x-2 ) | |
| = | (ex + x-1)-1 (ex - x-2) | ||