Derivatives of Constant Multiples

Example:
H = 2*((t + 1)2)-1

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 = 2 ((t + 1)2)-1

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 = 2 ((t + 1)2)-1
we can think of as
H = c f(t)
So the derivative is
H ' = ( c f(t) )'
  = c f '(t)  
  = 2 (((t + 1)2)-1)'  
and we just need to know the derivative on the right-hand side of the equation. In this case this is
((t + 1)2)-1 = 2*(-1)*((t + 1)2)-2 (t + 1) (1 + 0) (by the chain rule)
so the finished derivative is
H ' = 2 ( 2*(-1)*((t + 1)2)-2 (t + 1) (1 + 0) )
  = (-4)*((t + 1)2)-2 (t + 1)
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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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glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.