Derivatives of Constant Multiples

Example:
C(q) = (-4)*(3*q4 + 3)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:

C(q) = -4 (3*q4 + 3)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,

C(q) = -4 (3*q4 + 3)4
we can think of as
C(q) = c f(q)
So the derivative is
C '(q) = ( c f(q) )'
  = c f '(q)  
  = -4 ((3*q4 + 3)4)'  
and we just need to know the derivative on the right-hand side of the equation. In this case this is
(3*q4 + 3)4 = 4*(3*q4 + 3)3 (3*4*q3 + 0) (by the chain rule)
so the finished derivative is
C '(q) = -4 ( 4*(3*q4 + 3)3 (3*4*q3 + 0) )
  = (-192)*q3 (3*q4 + 3)3
[]


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.