Cite This Page
 
To Go
Computing Derivatives
Computing Derivatives
group rates for schools and districts
ADVERTISEMENT

Derivative of a Constant Multiple of a Function

Start with taking the derivatives of some lines.

Sample Problem

Find the derivative of each function.

  • f(x) = x
      
  • g(x) = 3f(x)
      
  • h(x) = 3g(x)

Answer. These are all lines.

  • The derivative of f(x) = x is the slope of the line f(x) = x, which is 1. Therefore f'(x) = 1.
      
  • Since f(x) = x,
      
    g(x) = 3f(x) = 3x.
      
    This is a line with slope 3, therefore g'(x) = 3.
      
  • Since g(x) = 3x,
      
    h(x) = 3g(x) = 3(3x) = 9x. 
    This is a line with slope 9, therefore h'(x) = 9.
      

If we multiply a function by 3, the derivative gets multiplied by 3 also. If we multiply a function by 2, the derivative gets multiplied by 2. And so on. The derivative of the function cf(x), assuming f is differentiable, is cf'(x).

In symbols, if g(x) = cf(x) where c is a constant, then 

g'(x) = cf'(x).

In words, if we have a function f and multiply it by some constant c to find a new function, then the derivative of that new function is c multiplied by the derivative of f. This works for any differentiable function f and any constant c.

In pictures, it's easiest to see what's going on with a line. If we take a line y = mx + b, it looks something like this:

If we multiply the whole line by 3, the line gets stretched vertically:

Now the line is 3 times steeper. Although it's a bit harder to see the picture with curvy functions, the idea is the same. If we stretch (or shrink) the function vertically, we're also stretching (or shrinking) its derivative.

Next Page: Multiplication by -1
Previous Page: Derivatives of More Complicated Functions

Need help with College?