Derivatives of Compositions

Example:
z = tan(s3 + 7)

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 = tan([])
  where  [] = s3 + 7

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 = tan([])
  where  [] = s3 + 7
we can think of as
z = f( [] ) , where  [] = g(s) = s3 + 7
So the derivative is
z ' = ( f( [] ) )'
  = f '( [] ) ( [] )'  
  = ( tan([]) )' ( s3 + 7 )'
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( tan([]) )' = 1 / (cos([]))2 (by the derivative rules for basic functions)
( s3 + 7 )' = ( 3*s2 + 0 ) (by the derivative rule for sums, power rule, and the derivative rule for constants)
so the finished derivative is
z ' = 1 / (cos([]))2 ( 3*s2 + 0 )
  = 1 / (cos(s3 + 7))2 ( 3*s2 + 0 )
  = 3*s2 (1 / (cos(s3 + 7))2)
[]


additional explanation for the chain rule
see another chain rule example
practice gateway test
previous page
Page Generated: Wed Jan 14 01:18:51 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.