Derivative of a Constant Multiple of a Function
Start with taking the derivatives of some lines.
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.