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:

V

=

()-1/4

where

=

9*x + sin(9)

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,

V

=

()-1/4

where

=

9*x + sin(9)

V

=

f( )

, where

=

g(x)

=

9*x + sin(9)

V '	= (	f( )		)'
	=	f '( )	( )'
	=	( ()-1/4 )'	( 9*x + sin(9) )'

( ()-1/4 )'	=	(-1/4)*()-5/4	(by the power rule)
( 9*x + sin(9) )'	=	( 9 + 0 )	(by the derivative rule for sums, rule for constant multiples, and the derivative rule for constants)

V '	=	(-1/4)*()-5/4	( 9 + 0 )
	=	(-1/4)*(9*x + sin(9))-5/4	( 9 + 0 )
	=	(-9/4)*(9*x + sin(9))-5/4