Derivatives of Sums

Example:
h(t) = t-(1/2) + 4*sin(sqrt(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:

h(t) = ( t-(1/2) ) + ( 4*sin(sqrt(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,

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


additional explanation for the derivative of sums
see another derivative of sums example
practice gateway test
previous page
Page Generated: Sat Jan 17 00:12:01 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.