Derivatives of Compositions

Example:
y = ln(ln(q) - q-1/4)

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  [] = ln(q) - q-1/4

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  [] = ln(q) - q-1/4
we can think of as
y = f( [] ) , where  [] = g(q) = ln(q) - q-1/4
So the derivative is
y ' = ( f( [] ) )'
  = f '( [] ) ( [] )'  
  = ( ln([]) )' ( ln(q) - q-1/4 )'
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)
( ln(q) - q-1/4 )' = ( q-1 - (-1/4)*q-5/4 ) (by the derivative rule for sums, derivative rules for basic functions, and the power rule)
so the finished derivative is
y ' = ([])-1 ( q-1 - (-1/4)*q-5/4 )
  = (ln(q) - q-1/4)-1 ( q-1 - (-1/4)*q-5/4 )
  = (ln(q) - q-1/4)-1 (q-1 + (1/4)*q-5/4)
[]


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.