Derivatives of Compositions

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)

=

()-p

where

=

sqrt(z)

The Chain Rule (Derivative Rule for Compositions):

The derivative of a composition is the product of the derivative of the outer function (with the inner function plugged in) and the derivative of the inner function.

If   z = f( g(x) ) then the derivative of z is   z' = ( f(g(x)) )'     = f '(g(x)) g '(x)

Or, if   z = f( ), where = g(x) then the derivative of z is   z' = ( f( ) )'     = f '( ) ( )'     = f '( ) g '(x)

So our example,

B(z)

=

()-p

where

=

sqrt(z)

B(z)

=

f( )

, where

=

g(z)

=

sqrt(z)

B '(z)	= (	f( )		)'
	=	f '( )	( )'
	=	( ()-p )'	( sqrt(z) )'

( ()-p )'	=	(-p)*()-p-1	(by the power rule)
( sqrt(z) )'	=	( (1/2)*z-1/2 )	(by the power rule, with exponent 1/2)

B '(z)	=	(-p)*()-p-1	( (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