Derivatives of Sums

Example:
r(s) = ((-5)*s)4 - 4*(3 - s)3

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:

r(s) = ( ((-5)*s)4 ) - ( 4*(3 - s)3 )

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,

r(s) = ( ((-5)*s)4 ) - ( 4*(3 - s)3 )
we can think of as
r(s) = f(s) - g(s)
So the derivative is
r '(s) = ( f(s) - g(s) )'
  = f '(s) - g '(s)  
  = ( ((-5)*s)4) ' - ( 4*(3 - s)3) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( ((-5)*s)4 )' = (-5)*4*((-5)*s)3 (by the chain rule)
( 4*(3 - s)3 )' = 4*3*(3 - s)2 (0 - 1) (by the rule for constant multiples, and the chain rule)
so the finished derivative is
r '(s) = (-5)*4*((-5)*s)3 - 4*3*(3 - s)2 (0 - 1)  
  = (-20)*((-5)*s)3 + 12*(3 - s)2
[]

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.