Derivatives of Constant Multiples

Example:
y = 5*sin(sqrt(t))

The first thing to notice when finding the derivative of this function is that it is the product of a constant and another function, as shown in color below:

y = 5 sin(sqrt(t))

The Derivative Rule for Constant Multiples:

The derivative of a constant multiple is the constant times thederivative of the function.
If
  z = c ( f(x) )
then the derivative of z is
  z' = ( c f(x) )'
    = c f '(x)

So our example,

y = 5 sin(sqrt(t))
we can think of as
y = c f(t)
So the derivative is
y ' = ( c f(t) )'
  = c f '(t)  
  = 5 (sin(sqrt(t)))'  
and we just need to know the derivative on the right-hand side of the equation. In this case this is
sin(sqrt(t)) = (1/2)*cos(sqrt(t)) t-1/2 (by the chain rule)
so the finished derivative is
y ' = 5 ( (1/2)*cos(sqrt(t)) t-1/2 )
  = (5/2)*t-1/2 cos(sqrt(t))
[]


additional explanation for the derivative of constant multiples
see another derivative of constant multiples example
practice gateway test
previous page
Page Generated: Wed Jan 7 09:11:16 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.