Derivatives of Sums

Example:
x(t) = ((-5)*t)3 + 7 + 4*t - p(ln(3))

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:

x(t) = ( ((-5)*t)3 ) + ( 7 ) + ( 4*t ) - ( p(ln(3)) )

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,

x(t) = ( ((-5)*t)3 ) + ( 7 ) + ( 4*t ) - ( p(ln(3)) )
we can think of as
x(t) = f(t) + g(t) + h(t) - p(t)
So the derivative is
x '(t) = ( f(t) + g(t) + h(t) - p(t) )'
  = f '(t) + g '(t) + h '(t) - p '(t)  
  = ( ((-5)*t)3) ' + ( 7) ' + ( 4*t) ' - ( p(ln(3))) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( ((-5)*t)3 )' = (-5)*3*((-5)*t)2 (by the chain rule)
( 7 )' = 0 (by the derivative rule for constants)
( 4*t )' = 4 (by the rule for constant multiples, and the derivative rule for variables)
( p(ln(3)) )' = 0 (by the derivative rule for constants)
so the finished derivative is
x '(t) = (-5)*3*((-5)*t)2 + 0 + 4 - 0  
  = (-15)*((-5)*t)2 + 4
[]

additional explanation for the derivative of sums
see another derivative of sums example
practice gateway test
previous page
Page Generated: Sun Nov 16 06:24:31 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.