Derivatives of Compositions

Example:
z = sin(tan(t) - t-p)

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

z = sin([])
  where  [] = tan(t) - t-p

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,

z = sin([])
  where  [] = tan(t) - t-p
we can think of as
z = f( [] ) , where  [] = g(t) = tan(t) - t-p
So the derivative is
z ' = ( f( [] ) )'
  = f '( [] ) ( [] )'  
  = ( sin([]) )' ( tan(t) - t-p )'
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( sin([]) )' = cos([]) (by the derivative rules for basic functions)
( tan(t) - t-p )' = ( 1 / (cos(t))2 - (-p)*t-p-1 ) (by the derivative rule for sums, derivative rules for basic functions, and the power rule)
so the finished derivative is
z ' = cos([]) ( 1 / (cos(t))2 - (-p)*t-p-1 )
  = cos(tan(t) - t-p) ( 1 / (cos(t))2 - (-p)*t-p-1 )
  = cos(tan(t) - t-p) (1 / (cos(t))2 - (-p)*t-p-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.