Derivatives of Sums

Example:
D = (5*p)4 + 7*p2 - (4*p - 3)2

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:

D = ( (5*p)4 ) + ( 7*p2 ) - ( (4*p - 3)2 )

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,

D = ( (5*p)4 ) + ( 7*p2 ) - ( (4*p - 3)2 )
we can think of as
D = f(p) + g(p) - h(p)
So the derivative is
D ' = ( f(p) + g(p) - h(p) )'
  = f '(p) + g '(p) - h '(p)  
  = ( (5*p)4) ' + ( 7*p2) ' - ( (4*p - 3)2) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( (5*p)4 )' = 5*4*(5*p)3 (by the chain rule)
( 7*p2 )' = 7*2*p (by the rule for constant multiples, and the power rule)
( (4*p - 3)2 )' = 2*(4*p - 3) (4 - 0) (by the chain rule)
so the finished derivative is
D ' = 5*4*(5*p)3 + 7*2*p - 2*(4*p - 3) (4 - 0)  
  = 20*(5*p)3 + 14*p - 8*(4*p - 3)
[]

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.