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:

y

=

()1/4

where

=

sin(t)

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,

y

=

()1/4

where

=

sin(t)

y

=

f( )

, where

=

g(t)

=

sin(t)

y '	= (	f( )		)'
	=	f '( )	( )'
	=	( ()1/4 )'	( sin(t) )'

( ()1/4 )'	=	(1/4)*()-3/4	(by the power rule)
( sin(t) )'	=	( cos(t) )	(by the derivative rules for basic functions)

y '	=	(1/4)*()-3/4	( cos(t) )
	=	(1/4)*(sin(t))-3/4	( cos(t) )
	=	(1/4)*(sin(t))-3/4 cos(t)