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:

h(x)

=

e

where

=

cos(x)

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,

h(x)

=

e

where

=

cos(x)

h(x)

=

f( )

, where

=

g(x)

=

cos(x)

h '(x)	= (	f( )		)'
	=	f '( )	( )'
	=	( e )'	( cos(x) )'

( e )'	=	e	(by the derivative rules for basic functions)
( cos(x) )'	=	( (-1)*sin(x) )	(by the derivative rules for basic functions)

h '(x)	=	e	( (-1)*sin(x) )
	=	ecos(x)	( (-1)*sin(x) )
	=	(-1)*ecos(x) sin(x)