The first thing to notice when finding the derivative of this function is that it is a composition of two functions, as shown below:
| B(z) | = | ( |
| where | |
= | sqrt(z) |
The Chain Rule (Derivative Rule for Compositions):
If
|
Or, if
|
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So our example,
| B(z) | = | ( |
| where | |
= | sqrt(z) |
| B(z) | = | f( |
, where | |
= | g(z) | = | sqrt(z) |
| B '(z) | = ( | f( |
)' | |
| = | f '( |
( |
||
| = | ( ( |
( sqrt(z) )' | ||
| ( ( |
= | (-p)*( |
(by the power rule) |
| ( sqrt(z) )' | = | ( (1/2)*z-1/2 ) | (by the power rule, with exponent 1/2) |
| B '(z) | = | (-p)*( |
( (1/2)*z-1/2 ) |
| = | (-p)*(sqrt(z))-p-1 | ( (1/2)*z-1/2 ) | |
| = | (1/2(-p))*z-1/2 (sqrt(z))-p-1 | ||