Derivatives of Sums

Example:
Z = (s + 4)7 - (p)*s5 + 7

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:

Z = ( (s + 4)7 ) - ( (p)*s5 ) + ( 7 )

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,

Z = ( (s + 4)7 ) - ( (p)*s5 ) + ( 7 )
we can think of as
Z = f(s) - g(s) + h(s)
So the derivative is
Z ' = ( f(s) - g(s) + h(s) )'
  = f '(s) - g '(s) + h '(s)  
  = ( (s + 4)7) ' - ( (p)*s5) ' + ( 7) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( (s + 4)7 )' = 7*(s + 4)6 (1 + 0) (by the chain rule)
( (p)*s5 )' = (p)*5*s4 (by the rule for constant multiples, and the power rule)
( 7 )' = 0 (by the derivative rule for constants)
so the finished derivative is
Z ' = 7*(s + 4)6 (1 + 0) - (p)*5*s4 + 0  
  = 7*(s + 4)6 - (5p)*s4
[]

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.