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:

C(x)

=

tan()

where

=

x3 - 8*x2 - 1

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,

C(x)

=

tan()

where

=

x3 - 8*x2 - 1

C(x)

=

f( )

, where

=

g(x)

=

x3 - 8*x2 - 1

C '(x)	= (	f( )		)'
	=	f '( )	( )'
	=	( tan() )'	( x3 - 8*x2 - 1 )'

( tan() )'	=	1 / (cos())2	(by the derivative rules for basic functions)
( x3 - 8*x2 - 1 )'	=	( 3*x2 - 8*2*x - 0 )	(by the derivative rule for sums, power rule, rule for constant multiples, and the derivative rule for constants)

C '(x)	=	1 / (cos())2	( 3*x2 - 8*2*x - 0 )
	=	1 / (cos(x3 - 8*x2 - 1))2	( 3*x2 - 8*2*x - 0 )
	=	(1 / (cos(x3 - 8*x2 - 1))2) (3*x2 - 16*x)