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(q)

=

ln()

where

=

(e)2 + 5*q

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(q)

=

ln()

where

=

(e)2 + 5*q

C(q)

=

f( )

, where

=

g(q)

=

(e)2 + 5*q

C '(q)	= (	f( )		)'
	=	f '( )	( )'
	=	( ln() )'	( (e)2 + 5*q )'

( ln() )'	=	()-1	(by the derivative rules for basic functions)
( (e)2 + 5*q )'	=	( 0 + 5 )	(by the derivative rule for sums, derivative rule for constants, and the rule for constant multiples)

C '(q)	=	()-1	( 0 + 5 )
	=	((e)2 + 5*q)-1	( 0 + 5 )
	=	5*((e)2 + 5*q)-1