Derivatives of Constant Multiples

Example:
f(x) = (2/3)*cos(sqrt(x) - 2*x)

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:

f(x) = 2/3 cos(sqrt(x) - 2*x)

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,

f(x) = 2/3 cos(sqrt(x) - 2*x)
we can think of as
f(x) = c g(x)
So the derivative is
f '(x) = ( c g(x) )'
  = c g '(x)  
  = 2/3 (cos(sqrt(x) - 2*x))'  
and we just need to know the derivative on the right-hand side of the equation. In this case this is
cos(sqrt(x) - 2*x) = (-1)*sin(sqrt(x) - 2*x) ((1/2)*x-1/2 - 2) (by the chain rule)
so the finished derivative is
f '(x) = 2/3 ( (-1)*sin(sqrt(x) - 2*x) ((1/2)*x-1/2 - 2) )
  = (-2/3)*sin(sqrt(x) - 2*x) ((1/2)*x-1/2 - 2)
[]


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.