Derivatives of Sums

Example:
y = x3 + x2 - 4*x

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 = ( x3 ) + ( x2 ) - ( 4*x )

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 = ( x3 ) + ( x2 ) - ( 4*x )
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)  
  = ( x3) ' + ( x2) ' - ( 4*x) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( x3 )' = 3*x2 (by the power rule)
( x2 )' = 2*x (by the power rule)
( 4*x )' = 4 (by the rule for constant multiples, and the derivative rule for variables)
so the finished derivative is
y ' = 3*x2 + 2*x - 4  
  = 3*x2 + 2*x - 4
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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.