Derivatives of Sums

Example:
L(q) = 10*(tan(q))4 + 6

The first thing to notice when finding the derivative of this function is that it is the sum of several terms, as shown in color below:

L(q) = ( 10*(tan(q))4 ) + ( 6 )

The Derivative Rule for Sums:

The derivative of a sum is the sum of the derivatives.
If
  z = ( f(x) + g(x) )
then the derivative of z is
  z' = ( f(x) + g(x) )'
    = f '(x) + g'(x)

So our example,

L(q) = ( 10*(tan(q))4 ) + ( 6 )
we can think of as
L(q) = f(q) + g(q)
So the derivative is
L '(q) = ( f(q) + g(q) )'
  = f '(q) + g '(q)  
  = ( 10*(tan(q))4) ' + ( 6) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( 10*(tan(q))4 )' = 10*4*(tan(q))3 (1 / (cos(q))2) (by the rule for constant multiples, and the chain rule)
( 6 )' = 0 (by the derivative rule for constants)
so the finished derivative is
L '(q) = 10*4*(tan(q))3 (1 / (cos(q))2) + 0  
  = 40*(tan(q))3 (1 / (cos(q))2)
[]

additional explanation for the derivative of sums
see another derivative of sums example
practice gateway test
previous page
Page Generated: Mon Feb 2 05:11:16 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.