# At a Glance - Derivative of a Constant Multiple of a Function

We'll start with taking the derivatives of some lines.

### Sample Problem

Find the derivative of each function.

*f*(*x*) =*x*

*g*(*x*) = 3*f*(*x*)

*h*(*x*) = 3*g*(*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*) = 3*f*(*x*) = 3*x*.

This is a line with slope 3, therefore*g '*(*x*) = 3.

- Since
*g*(*x*) = 3*x*,

*h*(*x*) = 3*g*(*x*) = 3(3*x*) = 9*x.*

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.

#### Example 1

Let |

#### Example 2

Let |

#### Exercise 1

Find the derivative of the function.

*f*(*x*) =*x*

#### Exercise 2

What's the derivative of the following function?

*g*(*x*) = 2*f*(*x*)

#### Exercise 3

Find the derivative of the function.

*h*(*x*) = 2*g*(*x*)

#### Exercise 4

Find the derivative of the function.

*k*(*x*) = 2*h*(*x*)

#### Exercise 5

Find the derivative of the function.

*f*(*x*) = 5*x*^{3}

#### Exercise 6

Find the derivative of the function.

*f*(*x*) = 6cos*x*

#### Exercise 7

Find the derivative of the function.

*f*(*x*) = 10sin*x*

#### Exercise 8

Find the derivative of the function.

*g*(*x*) = 6*e*^{x}

#### Exercise 9

Find the derivative of the function.

*g*(*x*) = 4(5)^{x}

#### Exercise 10

Find the derivative of the function.

#### Exercise 11

Find the derivative of the function.

*h*(*x*) = 9 ln*x*

#### Exercise 12

Find the derivative of the function.

#### Exercise 13

Find the derivative of the function.

*h*(*x*) = 4*x*^{12}