Derivatives of Constant Multiples

Example:
y = A*sin(B*x + C)

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 = A sin(B*x + C)

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 = A sin(B*x + C)
we can think of as
y = c f(x)
So the derivative is
y ' = ( c f(x) )'
  = c f '(x)  
  = A (sin(B*x + C))'  
and we just need to know the derivative on the right-hand side of the equation. In this case this is
sin(B*x + C) = cos(B*x + C) (B + 0) (by the chain rule)
so the finished derivative is
y ' = A ( cos(B*x + C) (B + 0) )
  = (AB)*cos(B*x + C)
[]


additional explanation for the derivative of constant multiples
see another derivative of constant multiples example
practice gateway test
previous page
Page Generated: Mon Jan 19 10:01:53 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.