Derivatives of Constant Multiples

Example:
y = (p)*ln(5*x2 + p)

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:

y = p ln(5*x2 + p)

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,

y = p ln(5*x2 + p)
we can think of as
y = c f(x)
So the derivative is
y ' = ( c f(x) )'
  = c f '(x)  
  = p (ln(5*x2 + p))'  
and we just need to know the derivative on the right-hand side of the equation. In this case this is
ln(5*x2 + p) = (5*x2 + p)-1 (5*2*x + 0) (by the chain rule)
so the finished derivative is
y ' = p ( (5*x2 + p)-1 (5*2*x + 0) )
  = (10p)*x (5*x2 + p)-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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Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.