Derivatives of Sums

Example:
L(p) = (-1)*p4 - 2*p + p

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(p) = ( (-1)*p4 ) - ( 2*p ) + ( p )

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(p) = ( (-1)*p4 ) - ( 2*p ) + ( p )
we can think of as
L(p) = f(p) - g(p) + h(p)
So the derivative is
L '(p) = ( f(p) - g(p) + h(p) )'
  = f '(p) - g '(p) + h '(p)  
  = ( (-1)*p4) ' - ( 2*p) ' + ( p) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( (-1)*p4 )' = (-1)*4*p3 (by the rule for constant multiples, and the power rule)
( 2*p )' = 2 (by the rule for constant multiples, and the derivative rule for variables)
( p )' = 0 (by the derivative rule for constants)
so the finished derivative is
L '(p) = (-1)*4*p3 - 2 + 0  
  = (-4)*p3 - 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.