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

=

ln()

where

=

4*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

=

ln()

where

=

4*t

y

=

f( )

, where

=

g(t)

=

4*t

y '	= (	f( )		)'
	=	f '( )	( )'
	=	( ln() )'	( 4*t )'

( ln() )'	=	()-1	(by the derivative rules for basic functions)
( 4*t )'	=	( 4 )	(by the rule for constant multiples, and the derivative rule for variables)

y '	=	()-1	( 4 )
	=	(4*t)-1	( 4 )
	=	4*(4*t)-1