Derivatives of Compositions

Example:

g(y) = ln(e^y - y^e)

The first thing to notice when finding the derivative of this function is that it is a composition of two functions, as shown below:

g(y)

ln(

)

where

e^y - y^e

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.

f( g(x) )

then the derivative of z is

	z'	=	( f(g(x)) )'
		=	f '(g(x))	g '(x)

Or, if

), where

= g(x)

then the derivative of z is

z'	=	( f( ) )'
	=	f '( )	( )'
	=	f '( )	g '(x)

So our example,

g(y)

ln(

)

where

e^y - y^e

we can think of as

g(y)

)

, where

g(y)

e^y - y^e

So the derivative is

g '(y)	= (	f( )		)'
	=	f '( )	( )'
	=	( ln() )'	( e^y - y^e )'

and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are

( ln() )'	=	()^-1	(by the derivative rules for basic functions)
( e^y - y^e )'	=	( e^y - e*y^e-1 )	(by the derivative rule for sums, derivative rules for basic functions, and the power rule)

so the finished derivative is

g '(y)	=	()^-1	( e^y - e*y^e-1 )
	=	(e^y - y^e)^-1	( e^y - e*y^e-1 )
	=	(e^y - y^e)^-1 (e^y - e*y^e-1)

additional explanation for the chain rule
see another chain rule example
practice gateway test
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Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.