Derivatives of Constant Multiples

Example:
f(x) = 18*(sin(x))4

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:

f(x) = 18 (sin(x))4

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,

f(x) = 18 (sin(x))4
we can think of as
f(x) = c g(x)
So the derivative is
f '(x) = ( c g(x) )'
  = c g '(x)  
  = 18 ((sin(x))4)'  
and we just need to know the derivative on the right-hand side of the equation. In this case this is
(sin(x))4 = 4*(sin(x))3 cos(x) (by the chain rule)
so the finished derivative is
f '(x) = 18 ( 4*(sin(x))3 cos(x) )
  = 72*(sin(x))3 cos(x)
[]


additional explanation for the derivative of constant multiples
see another derivative of constant multiples example
practice gateway test
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Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.