Derivatives of Constant Multiples

Example:
y = 9*sin(sqrt(q) - 10*q)

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 = 9 sin(sqrt(q) - 10*q)

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 = 9 sin(sqrt(q) - 10*q)
we can think of as
y = c f(q)
So the derivative is
y ' = ( c f(q) )'
  = c f '(q)  
  = 9 (sin(sqrt(q) - 10*q))'  
and we just need to know the derivative on the right-hand side of the equation. In this case this is
sin(sqrt(q) - 10*q) = cos(sqrt(q) - 10*q) ((1/2)*q-1/2 - 10) (by the chain rule)
so the finished derivative is
y ' = 9 ( cos(sqrt(q) - 10*q) ((1/2)*q-1/2 - 10) )
  = 9*cos(sqrt(q) - 10*q) ((1/2)*q-1/2 - 10)
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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.