Derivatives of Sums

Example:
y = ((-3)*x + 5)4 - (p)*x3 + 1

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 = ( ((-3)*x + 5)4 ) - ( (p)*x3 ) + ( 1 )

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 = ( ((-3)*x + 5)4 ) - ( (p)*x3 ) + ( 1 )
we can think of as
y = f(x) - g(x) + h(x)
So the derivative is
y ' = ( f(x) - g(x) + h(x) )'
  = f '(x) - g '(x) + h '(x)  
  = ( ((-3)*x + 5)4) ' - ( (p)*x3) ' + ( 1) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( ((-3)*x + 5)4 )' = 4*((-3)*x + 5)3 (-3 + 0) (by the chain rule)
( (p)*x3 )' = (p)*3*x2 (by the rule for constant multiples, and the power rule)
( 1 )' = 0 (by the derivative rule for constants)
so the finished derivative is
y ' = 4*((-3)*x + 5)3 (-3 + 0) - (p)*3*x2 + 0  
  = (-12)*((-3)*x + 5)3 - (3p)*x2
[]

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.