Derivatives of Compositions

Example:
V = (tan(x))-6

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

V = ([])-6
  where  [] = tan(x)

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,

V = ([])-6
  where  [] = tan(x)
we can think of as
V = f( [] ) , where  [] = g(x) = tan(x)
So the derivative is
V ' = ( f( [] ) )'
  = f '( [] ) ( [] )'  
  = ( ([])-6 )' ( tan(x) )'
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( ([])-6 )' = (-6)*([])-7 (by the power rule)
( tan(x) )' = ( 1 / (cos(x))2 ) (by the derivative rules for basic functions)
so the finished derivative is
V ' = (-6)*([])-7 ( 1 / (cos(x))2 )
  = (-6)*(tan(x))-7 ( 1 / (cos(x))2 )
  = (-6)*(tan(x))-7 (1 / (cos(x))2)
[]


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.