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:

W

=

cos()

where

=

sqrt(y) - y3/2

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,

W

=

cos()

where

=

sqrt(y) - y3/2

W

=

f( )

, where

=

g(y)

=

sqrt(y) - y3/2

W '	= (	f( )		)'
	=	f '( )	( )'
	=	( cos() )'	( sqrt(y) - y3/2 )'

( cos() )'	=	(-1)*sin()	(by the derivative rules for basic functions)
( sqrt(y) - y3/2 )'	=	( (1/2)*y-1/2 - (3/2)*y1/2 )	(by the derivative rule for sums, power rule, with exponent 1/2, and the power rule (again))

W '	=	(-1)*sin()	( (1/2)*y-1/2 - (3/2)*y1/2 )
	=	(-1)*sin(sqrt(y) - y3/2)	( (1/2)*y-1/2 - (3/2)*y1/2 )
	=	(-1)*sin(sqrt(y) - y3/2) ((1/2)*y-1/2 - (3/2)*y1/2)