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:

f(x)

=

sin()

where

=

sin(x) - x3

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,

f(x)

=

sin()

where

=

sin(x) - x3

f(x)

=

f( )

, where

=

g(x)

=

sin(x) - x3

f '(x)	= (	f( )		)'
	=	f '( )	( )'
	=	( sin() )'	( sin(x) - x3 )'

( sin() )'	=	cos()	(by the derivative rules for basic functions)
( sin(x) - x3 )'	=	( cos(x) - 3*x2 )	(by the derivative rule for sums, derivative rules for basic functions, and the power rule)

f '(x)	=	cos()	( cos(x) - 3*x2 )
	=	cos(sin(x) - x3)	( cos(x) - 3*x2 )
	=	cos(sin(x) - x3) (cos(x) - 3*x2)