Derivatives of Constant Multiples

Example:
L(t) = A*t5 (t2 - 2)3

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:

L(t) = A t5 (t2 - 2)3

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,

L(t) = A t5 (t2 - 2)3
we can think of as
L(t) = c f(t)
So the derivative is
L '(t) = ( c f(t) )'
  = c f '(t)  
  = A (t5 (t2 - 2)3)'  
and we just need to know the derivative on the right-hand side of the equation. In this case this is
t5 (t2 - 2)3 = 5*t4 (t2 - 2)3 + t5 3*(t2 - 2)2 (2*t - 0) (by the product rule, and the power rule, and chain rule)
so the finished derivative is
L '(t) = A ( 5*t4 (t2 - 2)3 + t5 3*(t2 - 2)2 (2*t - 0) )
  = A*(5*t4 (t2 - 2)3 + 6*t6 (t2 - 2)2)
[]

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.