# The Power Rule

We are often interested in making things simpler and more time-efficient in day-to-day life. Would you wash dishes by hand if you have a dishwasher? Or re-type a paragraph if you can copy and paste? We can say the same for mathematics!

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Take the Derivative as an example. Finding the derivative of a function is one of the basic operations in Calculus. However, using limits can be time-consuming as there are many steps and lots of algebra are involved. Here's an example of that process.

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

Use the definition of a derivative.

$\frac{df}{dx}=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}$

Evaluate $f\left(x+h\right)$ and $f\left(x\right)$.

$\frac{df}{dx}=\underset{h\to 0}{\mathrm{lim}}\frac{{\left(x+h\right)}^{2}-{x}^{2}}{h}$

Expand ${\left(x+h\right)}^{2}$.

$\frac{df}{dx}=\underset{h\to 0}{\mathrm{lim}}\frac{{x}^{2}+2xh+{h}^{2}-{x}^{2}}{h}$

Simplify.

$\frac{df}{dx}=\underset{h\to 0}{\mathrm{lim}}\left(2x+h\right)$

Evaluate the limit.

$\frac{df}{dx}=2x$

The derivative of $f\left(x\right)={x}^{2}$is$\frac{df}{dx}=2x$.

Lots of steps, right? Rather than doing all these procedures, there are many formulas in calculus that we can use to find derivatives with fewer steps, saving us time and mental energy. These formulas are known as derivative rules, and one of these derivative rules is the power rule.

## Power rule formula and examples

One of the primary functions found in calculus is the power function.

Power functions are functions where the variable is the base and is raised to any real number power.

$f\left(x\right)={x}^{n}$

These functions are essential in Calculus for building more advanced functions, like polynomial functions or rational functions. We can find the derivative of a power function using what is known as the power rule. Let's take a look at it.

The power rule is a formula for finding the derivative of a power function. Let $n$be a real number, then:

$\frac{d}{dx}{x}^{n}=n{x}^{n-1}$

This rule can make finding derivatives in calculus much simpler! Let's take a look at some examples.

Find the derivative of $f\left(x\right)={x}^{5}$.

Identify the power of the power function. This function has a power of 5.

$f\left(x\right)={x}^{{5}}$

Differentiate using The Power Rule.

$\frac{df}{dx}={5}{x}^{{5}-1}$

Simplify the exponent.

$\frac{df}{dx}=5{x}^{4}$

The derivative of $f\left(x\right)={x}^{5}$is$\frac{df}{dx}=5{x}^{4}$.

We can use the power rule in combination with other Differentiation Rules to find the derivative of a polynomial function. Let's look at an example of this process.

Find the derivative of $g\left(x\right)=3{x}^{4}-2{x}^{3}+x$.

Use the sum, difference, and constant multiplier rules.

$\frac{dg}{dx}=3\frac{d}{dx}{x}^{4}-2\frac{d}{dx}{x}^{3}+\frac{d}{dx}x$

Differentiate using the power rule.

$\frac{dg}{dx}=3\left(4{x}^{3}\right)-2\left(3{x}^{2}\right)+{x}^{0}$

Simplify.

$\frac{dg}{dx}=12{x}^{3}-6{x}^{2}+1$

The derivative of $g\left(x\right)=3{x}^{4}-2{x}^{3}+x$is$\frac{dg}{dx}=12{x}^{3}-6{x}^{2}+1$.

## Deriving the power rule

To prove the power rule, we will look at the derivative of $f\left(x\right)={x}^{n}$using limits. We need to find such a derivative using limits just once, proving our formula. Then we can use the formula whenever we need to differentiate a power function.

We begin by using the definition of a derivative.

$\frac{d}{dx}{x}^{n}=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}$

Next, evaluate $f\left(x+h\right)$and $f\left(x\right)$.

$\frac{d}{dx}{x}^{n}=\underset{h\to 0}{\mathrm{lim}}\frac{{\left(x+h\right)}^{n}-{x}^{n}}{h}$

We can use The Binomial Theorem to expand ${\left(x+h\right)}^{n}$.

${\left(x+h\right)}^{n}=\left(\begin{array}{c}n\\ 0\end{array}\right){x}^{n}+\left(\begin{array}{c}n\\ 1\end{array}\right){x}^{n-1}h+\left(\begin{array}{c}n\\ 2\end{array}\right){x}^{n-2}{h}^{2}+...+\left(\begin{array}{c}n\\ n\end{array}\right){h}^{n}$

The first two binomial coefficients are 1 and $n$, respectively.

${\left(x+h\right)}^{n}={x}^{n}+n{x}^{n-1}h+\left(\begin{array}{c}n\\ 2\end{array}\right){x}^{n-2}{h}^{2}+...+\left(\begin{array}{c}n\\ n\end{array}\right){h}^{n}$

To reflect the definition of a derivative, we need to subtract ${x}^{n}$and divide by $h$ on both sides of the equation.

We now have the following expression:

$\frac{{\left(x+h\right)}^{n}-{x}^{n}}{h}=n{x}^{n-1}+\left(\begin{array}{c}n\\ 2\end{array}\right){x}^{n-2}h+...+\left(\begin{array}{c}n\\ n\end{array}\right){h}^{n-1}$

As we take the limit as h goes to 0, every term that contains h vanishes. Hence, we are only left with $n{x}^{n-1}$.

$\underset{h\to 0}{\mathrm{lim}}\frac{{\left(x+h\right)}^{n}-{x}^{n}}{h}=n{x}^{n-1}$

Finally, we have arrived at the power rule.

$\frac{d}{dx}{x}^{n}=n{x}^{n-1}$

## Power rule for negative and fractional powers

We only proved the case involving positive integers. However, we can use the power rule when the powers are negative. The formula is the same.

Find the derivative of $g\left(x\right)={x}^{-3}$.

Identify the power of the power function. In this case, the power is -3.

$g\left(x\right)={x}^{{-}{3}}$

Differentiate using the power rule.

$\frac{dg}{dx}={-}{3}{x}^{{-}{3}-1}$

Simplify the exponent.

$\frac{dg}{dx}=-3{x}^{-4}$

We can also use the power rule for fractional powers, like in the case of a square root function.

Find the derivative of $h\left(x\right)=\sqrt{x}$.

Write the root as a fractional power.

$h\left(x\right)={x}^{\frac{1}{2}}$

Differentiate using the power rule.

$\frac{dh}{dx}=\frac{1}{2}{x}^{\frac{1}{2}-1}$

Simplify the power.

$\frac{dh}{dx}=\frac{1}{2}{x}^{-\frac{1}{2}}$

Write the negative power in the denominator.

Write the power as a root.

$\frac{dh}{dx}=\frac{1}{2}\frac{1}{\sqrt{x}}$

The power rule works when n is any real number. Luckily, the formula is the same for every case!

## More examples of the power rule

Calculus is full of different functions to which we can apply the rules of differentiation. In this section, we will look at more examples of derivatives using the power rule.

Find the derivative of $f\left(x\right)=2{x}^{4}-{x}^{2}$.

Use the sum, difference, and constant multiplier rules.

$\frac{df}{dx}=2\frac{d}{dx}{x}^{4}-\frac{d}{dx}{x}^{2}$

Differentiate using the power rule.

$\frac{df}{dx}=2\left(4{x}^{3}\right)-2x$

Simplify.

$\frac{df}{dx}=8{x}^{3}-2x$

The following example considers negative powers.

Find the derivative of $g\left(x\right)={x}^{2}+\frac{1}{{x}^{2}}$.

Write the power in the denominator as a negative power.

$g\left(x\right)={x}^{2}+{x}^{-2}$

Use the sum rule.

$\frac{dg}{dx}=\frac{d}{dx}\left({x}^{2}\right)+\frac{d}{dx}\left({x}^{-2}\right)$

Differentiate using The Power Rule.

$\frac{dg}{dx}=2x-2{x}^{-3}$

Write the negative exponent as a denominator.

$\frac{dg}{dx}=2x-\frac{2}{{x}^{3}}$

Let's look at more roots, which we can write as fractional powers.

Find the derivative of $h\left(x\right)=\sqrt[3]{x}+\frac{1}{{x}^{5}}$.

Write the root as a fractional power.

$h\left(x\right)={x}^{\frac{1}{3}}+\frac{1}{{x}^{5}}$

Write the power in the denominator as a negative power.

$h\left(x\right)={x}^{\frac{1}{3}}+{x}^{-5}$

Use the sum rule.

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

Differentiate using The Power Rule.

$\frac{dh}{dx}=\frac{1}{3}{x}^{\frac{1}{3}-1}-5{x}^{-5-1}$

Simplify the powers.

$\frac{dh}{dx}=\frac{1}{3}{x}^{-\frac{2}{3}}-5{x}^{-6}$

Write the negative powers as denominators.

Write the fractional powers as a power and a root.

$\frac{dh}{dx}=\frac{1}{3}\frac{1}{\sqrt[3]{{x}^{2}}}-\frac{5}{{x}^{6}}$

With enough practice, we can skip some of these steps.

### Common mistakes when using the power rule

We cannot use the power rule if the variable is the power of an expression.

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

One common mistake is applying the power rule to functions that are not power functions.

We cannot apply the power rule in a case like this because the given function is not a power function.

$\frac{d}{dx}{{2}}^{{x}}{\ne }{x}{{2}}^{x-1}$

Here we could use the Derivative of the Exponential Function instead.

Always remember to decrease the power by one after differentiating the function!

Find the derivative of $f\left(x\right)={x}^{5}$.

Another common mistake is forgetting to decrease the power of the power function.

$\frac{d}{dx}{{x}}^{{5}}{\ne }{5}{{x}}^{{5}}$

We must remember that the power drops when differentiating a power function.

$\frac{d}{dx}{x}^{5}=5{x}^{{4}}$

## The Power Rule - Key takeaways

• The power rule is a formula for finding the derivative of power functions.
• The formula for the power rule is as follows: $\frac{d}{dx}{x}^{n}=n{x}^{n-1}$
• We can use the power rule for any real number n, including negative numbers and fractions.
• We can use the power rule and basic derivative rules like the sum, difference, and constant multiplier rules to differentiate polynomial functions.

#### Flashcards in The Power Rule 5

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What is the power rule?

The power rule is a differentiation rule for finding the derivative of a power function.

When do you use the power rule?

You can use the power rule whenever you need to find the derivative of a power function. The power can be any real number.

How do you prove the power rule?

To prove the power rule you need to find the derivative of an arbitrary power function through limits. You will also need to use the binomial theorem to expand the power function.

## Test your knowledge with multiple choice flashcards

The Power Rule can be used if the power is a negative integer.

The Power Rule can be used if the power is a fraction.

The Power Rule can be used if the variable is the power.

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