# The Power Rule for Derivatives

## Introduction

Calculus is all about rates of change. To find a rate of change, we need to calculate a derivative. In this article, we're going to find out how to calculate derivatives for the simplest of all functions, the powers of $x$.

Let's start by thinking about a useful real world problem that you probably won't find in your maths textbook.

Sam decided to eat chips for lunch yesterday instead of the delicious anchovy, wheatgerm and broccoli sandwich that was in his lunch box. He hid his sandwich at the bottom of his school bag, and decided to calculate its rate of decomposition. After $t \text{ days}$, the amount (in grams) of sandwich remaining is given by the function:

$\text{sandwich}(t) = t^{-2}$.
What is the rate of decomposition of the sandwich after 2 days?

Sam needs to find the derivative of the function $\text{sandwich}(t) = t^{-2}$, and evaluate it at $t = 2$.

Sounds easy enough, but how do we find the derivative of $\text{sandwich}(t)$?

## The Power Rule

Sam's function $\text{sandwich}(t) = t^{-2}$ involves a power of $t$. There's a differentiation law that allows us to calculate the derivatives of powers of $t$, or powers of $x$, or powers of elephants, or powers of anything you care to think of. Strangely enough, it's called the Power Rule.

### So what does the power rule say?

The derivative of $x^n$ is $nx^{n-1}$

There are two common ways to write the derivative of a function

• If our function is $f(x)$, then we can write its derivative as $f'(x)$ using a little ' after the $f$. We pronounce $f'(x)$ as "f-dash of x" or "f-prime"
• If $y$ is our function of $x$, then we can also write its derivative as $\dfrac{dy}{dx}$ and call it "dee-y dee-x".

Now let's differentiate a few functions

#### Example

If $y = x^3$, what is $\dfrac{dy}{dx}$?

\begin{align*}\dfrac{dy}{dx} &= 3x^{3-1}\\ &= 3x^2\end{align*}

That wasn't too bad, was it? Let's try another one.

#### Example

Find the derivative of $f(x) = x^5$.

\begin{align*}f'(x) &= 5x^{5-1}\\ &= 5x^4\end{align*}

Let's try a slightly trickier example:

#### Example

Find the derivative of $f(x) = \dfrac{1}{x^2}$.

Note: You can use your index laws to write $\dfrac{1}{x^2} = x^{-2}$. Then you just apply the power rule as usual.

\begin{align*}f'(x) &= \dfrac{d}{dx}(x^{-2})\\ &= (-2)x^{(-2 - 1)}\\ &= -2x^{-3}\\ &= \dfrac{-2}{x^3}\end{align*}

The power rule works for fractional indices as well!

#### Example

Find the derivative of $f(x) = \sqrt{x}$.

Note: You can use your index laws to write $\sqrt{x} = x^{\frac{1}{2}}$. Then you just apply the power rule as usual.

\begin{align*}f'(x) &= \dfrac{d}{dx}(x^{\frac{1}{2}})\\ &= \dfrac{1}{2}\;x^{(\frac{1}{2} - 1)}\\ &= \dfrac{1}{2}\;x^{-\frac{1}{2}}\\ &= \dfrac{1}{2 \sqrt{x}}\end{align*}

## Patterns in the derivatives

Let's look at a table of derivatives of powers of $x$ and see if we can spot a pattern:

$y = x^n$ $nx^{n-1}$ $\dfrac{dy}{dx}$
$x$ $1x^{(1-1)} = x^0 = 1$ $1$
$x^2$ $2x^{2-1} = 2x^1 = 2x$ $2x$
$x^3$ $3x^{3 - 1} = 3x^2$ $3x^2$
$x^4$ $4x^{4 - 1} = 4x^3$ $4x^3$
... ... ...

The coefficients and powers of the derivatives step up by $1$ as $n$ steps up by $1$.

The same thing happens for negative powers:

$y = x^n$ $nx^{n-1}$ $\dfrac{dy}{dx}$
$x^{-1}$ $(-1)x^{(-1-1)} = -x^{-2}$ $-x^{-2}$
$x^{-2}$ $-2x^{-2-1} = -2x^{-3}$ $-2x^{-3}$
$x^{-3}$ $-3x^{-3 - 1} = -3x^{-4}$ $-3x^{-4}$
... ... ...

## Solving Sam's Problem

Now we know enough to solve Sam's problem. Sam's function was:

$\text{sandwich}(t) = t^{-2}$,

and he wants to know his sandwich's rate of decomposition after 2 days. So he needs to find the derivative of the sandwich function (crumbs?) and plug in $t = 2$. It looks like a job for the power rule!

Let's differentiate:

$\text{sandwich}'(t) = (-2)t^{-2 - 1} = -2t^{-3}$.

At $t = 2$:

$\text{sandwich}'(2) = (-2)t^{-2 - 1} = -2(2)^{-3} = \dfrac{-2}{8} = -\dfrac{1}{4}$.

So the sandwich is decomposing at a rate of $\dfrac{1}{4}$ grams per day.

You know what, Sam? I think it might be better for everyone if you throw that sandwich in the bin!

### Description

Calculus is the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two main types are differential calculus and integral calculus.

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