  Calculus

# What Is the Power Rule?

## Drew Zemke

Subject Matter Expert

The Power Rule is one of the fundamental derivative rules in the field of Calculus. In this article, we'll first discuss its definition and how to use it, and then take a deeper dive by looking at its application to a number of specific functions.

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### What Is the Power Rule?

The Power Rule is one of the first derivative rules that we come across when we’re learning about derivatives. It gives us a quick way to differentiate—that is, to take the derivative of—functions like $x^2$ and $x^3$, and since functions like that are ubiquitous throughout calculus, we use it frequently.

Combining the Power Rule with the rules for differentiating sums and constant multiples of functions, we can differentiate a polynomial function like $2x^3 - 5x^2 + x - 1$ without too much hassle. Additionally, we’ll see later that we can even use the Power Rule to differentiate some functions that aren’t written explicitly as powers of $x$, like $\sqrt{x}$ and $\frac{1}{x^3}$. Understanding the power rule opens up a decently sized library of functions that you can start using to explore the more conceptual or applied sides of derivatives as well.

### How to Use the Power Rule

Here’s the Power Rule in its most general form:  Based on that formula, we can break down the process for using the Power Rule into three steps:

1. Write the function that you want to differentiate with the Power Rule in the standard form of $f(x)=x^k$. (Sometimes this has already been done for you.) Make sure you know what $k$ is before computing!

2. To form the derivative, write the power in the original function ($k$), and then

3. Next to that, write the independent variable ($x$ in this case) with a power that is one less than it was before.  Let’s see how this works with a few examples.

• To differentiate $f(x)=x^7$, we don’t have to do anything for the first step since it’s already in the form of a power function. For the derivative, we write down the initial power ($7$) as a coefficient, then write $x$ with a decreased power:

$f’(x)=7x^{7-1} = 7x^6$

• Next, let’s compute the derivative of $g(x) = \sqrt{x}$. We first need to rewrite the function as a power function: $g(x) = x^{\frac{1}{2}}$. In this case, $k=\frac{1}{2}$, so that will be the number that we write as a coefficient and decrease by one to obtain the new power:

$g’(x) = \frac{1}{2} x^{\frac{1}{2} - 1} = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2} \frac{1}{\sqrt{x}}$

(That last simplification step is optional, but you’ll often see this particular derivative written that way.)

• Using slightly different notation and a different variable from  the previous examples, let’s compute $\frac{d}{da} \frac{1}{a^3}$. As in the previous example, we need to first write the function we’re differentiating as $a$ to $a^{-3}$. We can then apply the Power Rule as we have previously, using $k=-3$:

$\frac{d}{da} a^{-3} = (-3)a^{(-3)-1} = -3a^{-4}$

• As a final example, we can use the Power Rule in conjunction with the sum and constant multiple rules for derivatives to differentiate polynomials:

$\frac{d}{dx} (2x^3 - 5x^2 + x - 1)$

= $2 \frac{d}{dx} x^3 - 5 \frac{d}{dx} x^2 + \frac{d}{dx} x - \frac{d}{dx} 1$

= $2(3x^2) - 5(2x) + 1x^0 - 0$

= $6x^2 - 10x + 1$

### More Examples of the Power Rule

Finally, it can be useful to see how the Power Rule works on a sequence of similar functions. In the following table we’ve used the Power Rule to differentiate the first several (whole number) powers of $x$:  Here’s a similar table for negative integer powers of $x$.  You don’t need to memorize these tables in order to be successful in calculus; it’s more important to internalize the process behind applying the Power Rule. But on the other hand, studying the patterns in the derivatives of those functions may help you build a better understanding of how the Power Rule works.

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