The derivative of a constant (a number)

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

the-derivative-of-a-constant-or-number

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.

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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.

graph-of-x^2

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

graph-of-a-constant-function

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.

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Summary

You will probably never have trouble finding the derivative of a constant when it is part of a polynomial or other function. But, be careful at paying attention to the different forms a constant may take, as professors and teachers love checking if you notice things like that. Further, you can use this easy idea to help you remember the concept of the derivative as the slope at a point – something that you will work with even when the derivatives are much more complicated.

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