Derivatives of Constant Multiples

Example:
h(t) = (-3)*(2*t4 + 7)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:

h(t) = -3 (2*t4 + 7)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,

h(t) = -3 (2*t4 + 7)4
we can think of as
h(t) = c f(t)
So the derivative is
h '(t) = ( c f(t) )'
  = c f '(t)  
  = -3 ((2*t4 + 7)4)'  
and we just need to know the derivative on the right-hand side of the equation. In this case this is
(2*t4 + 7)4 = 4*(2*t4 + 7)3 (2*4*t3 + 0) (by the chain rule)
so the finished derivative is
h '(t) = -3 ( 4*(2*t4 + 7)3 (2*4*t3 + 0) )
  = (-96)*t3 (2*t4 + 7)3
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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.