Derivatives of Compositions

Example:
C(q) = ln((e)2 + 5*q)

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
we can think of as
C(q) = f( [] ) , where  [] = g(q) = (e)2 + 5*q
So the derivative is
C '(q) = ( f( [] ) )'
  = f '( [] ) ( [] )'  
  = ( ln([]) )' ( (e)2 + 5*q )'
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)2 + 5*q )' = ( 0 + 5 ) (by the derivative rule for sums, derivative rule for constants, and the rule for constant multiples)
so the finished derivative is
C '(q) = ([])-1 ( 0 + 5 )
  = ((e)2 + 5*q)-1 ( 0 + 5 )
  = 5*((e)2 + 5*q)-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.