Derivatives of Sums

Example:
y = tan(q) - (tan(q))1/8

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:

y = ( tan(q) ) - ( (tan(q))1/8 )

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,

y = ( tan(q) ) - ( (tan(q))1/8 )
we can think of as
y = f(q) - g(q)
So the derivative is
y ' = ( f(q) - g(q) )'
  = f '(q) - g '(q)  
  = ( tan(q)) ' - ( (tan(q))1/8) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( tan(q) )' = 1 / (cos(q))2 (by the derivative rules for basic functions)
( (tan(q))1/8 )' = (1/8)*(tan(q))-7/8 (1 / (cos(q))2) (by the chain rule)
so the finished derivative is
y ' = 1 / (cos(q))2 - (1/8)*(tan(q))-7/8 (1 / (cos(q))2)  
  = 1 / (cos(q))2 - (1/8)*(tan(q))-7/8 (1 / (cos(q))2)
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additional explanation for the derivative of sums
see another derivative of sums example
practice gateway test
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Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.