Derivatives of Sums

Example:
g(t) = 20*t3 + 5(p2) - 7*t

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:

g(t) = ( 20*t3 ) + ( 5(p2) ) - ( 7*t )

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,

g(t) = ( 20*t3 ) + ( 5(p2) ) - ( 7*t )
we can think of as
g(t) = f(t) + h(t) - p(t)
So the derivative is
g '(t) = ( f(t) + h(t) - p(t) )'
  = f '(t) + h '(t) - p '(t)  
  = ( 20*t3) ' + ( 5(p2)) ' - ( 7*t) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( 20*t3 )' = 20*3*t2 (by the rule for constant multiples, and the power rule)
( 5(p2) )' = 0 (by the derivative rule for constants)
( 7*t )' = 7 (by the rule for constant multiples, and the derivative rule for variables)
so the finished derivative is
g '(t) = 20*3*t2 + 0 - 7  
  = 60*t2 - 7
[]

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.