Derivatives of Compositions

Example:
g(y) = ln(ey - ye)

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  [] = ey - ye

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,

g(y) = ln([])
  where  [] = ey - ye
we can think of as
g(y) = f( [] ) , where  [] = g(y) = ey - ye
So the derivative is
g '(y) = ( f( [] ) )'
  = f '( [] ) ( [] )'  
  = ( ln([]) )' ( ey - ye )'
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)
( ey - ye )' = ( ey - e*ye-1 ) (by the derivative rule for sums, derivative rules for basic functions, and the power rule)
so the finished derivative is
g '(y) = ([])-1 ( ey - e*ye-1 )
  = (ey - ye)-1 ( ey - e*ye-1 )
  = (ey - ye)-1 (ey - e*ye-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.