Derivatives of Sums

Example:
H = 2*((t + 1)2)-1 + 3*(2*t)-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:

H = ( 2*((t + 1)2)-1 ) + ( 3*(2*t)-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,

H = ( 2*((t + 1)2)-1 ) + ( 3*(2*t)-1 )
we can think of as
H = f(t) + g(t)
So the derivative is
H ' = ( f(t) + g(t) )'
  = f '(t) + g '(t)  
  = ( 2*((t + 1)2)-1) ' + ( 3*(2*t)-1) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( 2*((t + 1)2)-1 )' = 2*2*(-1)*((t + 1)2)-2 (t + 1) (1 + 0) (by the rule for constant multiples, and the chain rule)
( 3*(2*t)-1 )' = 3*2*(-1)*(2*t)-2 (by the rule for constant multiples, and the chain rule)
so the finished derivative is
H ' = 2*2*(-1)*((t + 1)2)-2 (t + 1) (1 + 0) + 3*2*(-1)*(2*t)-2  
  = (-4)*((t + 1)2)-2 (t + 1) - 6*(2*t)-2
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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.