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

=

()-6

where

=

tan(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,

V

=

()-6

where

=

tan(x)

V

=

f( )

, where

=

g(x)

=

tan(x)

V '	= (	f( )		)'
	=	f '( )	( )'
	=	( ()-6 )'	( tan(x) )'

( ()-6 )'	=	(-6)*()-7	(by the power rule)
( tan(x) )'	=	( 1 / (cos(x))2 )	(by the derivative rules for basic functions)

V '	=	(-6)*()-7	( 1 / (cos(x))2 )
	=	(-6)*(tan(x))-7	( 1 / (cos(x))2 )
	=	(-6)*(tan(x))-7 (1 / (cos(x))2)