# The derivative of a constant (a number)

The derivative of any constant (which is just a way of saying any number), is zero.

This is easy enough to remember, but if you are a student currently taking calculus, you need to remember the many different forms a constant can take. First, let’s look at the more obvious cases.

### Example

Find the derivative of each function.

$$\text{(a) } f(x) = 1$$

$$\text{(b) } g(x) = 20$$

$$\text{(c) } k(x) = -\dfrac{117}{91}$$

### Solution

No matter how cute we try to get with crazy fractions, one fact remains: each of these are constants. Therefore, the derivative of each is zero.

$$\text{(a) } f^{\prime}(x) = \left(1\right)^{\prime} = 0$$

$$\text{(b) } g(x) = \left(20\right)^{\prime}=0$$

$$\text{(c) } k(x) = \left(-\dfrac{117}{91}\right)^{\prime}=0$$

## Constants in “disguise”

You learn about quite a few different types of constants in math. A couple that immediately come to mind are:

$$e \approx 2.718$$

$$\pi \approx 3.142$$

These are famous, but there are others that you have certainly worked with. Consider $$\sqrt{2}$$ or $$\ln\left(5\right)$$. Both of these are constants (if you aren’t sure, type them in your calculator – you will get the decimal equivalent) and so their derivatives are zero as well.

### Example

Find the derivative of each of the following.

$$\text{(a) } \pi^{3}$$

$$\text{(b) } \dfrac{\sqrt[3]{10}}{2}$$

$$\text{(c) } -(e-1)$$

### Solution

Again, each of these is a constant with derivative zero.

$$\text{(a) } \left(\pi^{3}\right)^{\prime}=0$$

$$\text{(b) } \left(\dfrac{\sqrt[3]{10}}{2}\right)^{\prime}=0$$

$$\text{(c) } \left(-(e-1)\right)^{\prime}=0$$

Don’t be fooled though. If we put an $$x$$ or other variable with any of these, they are no longer constants and the rules for finding their derivative is different. For example, $$\left( e^x \right)^{\prime} = e^x$$, not zero.

## Why is the derivative of a constant zero?

One way of thinking about the derivative, is as the slope of a function at a given point. With functions like $$f(x) = x^2$$ (graphed below), the slope can change from point to point because the graph is curved.

But what does the function look like if it is a constant function? Below is the graph of $$f(x) = 2.5$$.

This graph is a line, so the slope is the same at every point. Further, it is a horizontal line. The slope of any horizontal line is zero. Since the graph of any constant function is a horizontal line like this, the derivative is always zero.