Derivatives of Constant Multiples

Example:
P(t) = Q*ek*t

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:

P(t) = Q ek*t

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,

P(t) = Q ek*t
we can think of as
P(t) = c f(t)
So the derivative is
P '(t) = ( c f(t) )'
  = c f '(t)  
  = Q (ek*t)'  
and we just need to know the derivative on the right-hand side of the equation. In this case this is
ek*t = k*ek*t (by the chain rule)
so the finished derivative is
P '(t) = Q ( k*ek*t )
  = (Qk)*ek*t
[]


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.