# The Power Rule for Derivatives: What is the Power Rule?

## What is the Power Rule?

In calculus, what is the power rule? The power rule, which is also called the exponent rule, is a rule that tells the derivative of a **power function** of the form {eq}f(x)=ax^n {/eq} for {eq}a, x \in \mathbb{R} {/eq} and {eq}a, n \neq 0 {/eq}. Now, a **power function** is any function of the form above; that is, it is a function of the form of x raised to a fixed power n and multiplied by some coefficient a. Here are some examples of power functions:

- {eq}f(x)=\pi*x^2 {/eq}

- {eq}f(x)=-x^5 {/eq}

- {eq}f(x)=3x^{-1} {/eq}

Notice that the powers of x in these examples can be both positive and negative and that the coefficients can also be both positive and negative.

Here is the graph of {eq}y=x^2 {/eq}. All power functions of positive even power ({eq}x^{2n} {/eq}) will have this general shape:

Similarly, power functions of positive odd power will have the same general shape. Below is the graph of {eq}y=x^3 {/eq} which will show the typical shape of a positive odd-powered power function ({eq}x^{2n+1} {/eq}):

Although power functions can look differently, they will always be of the forms above depending on if they are odd-powered or even-powered power functions.

Now, although power functions can appear differently when graphed, differentiating them involves a simple rule called the **power rule**. This rule will give the derivative for any power function (and later on, any sum of power functions as well as power functions of negative exponent).

The power rule formula for a fundamental power function is:

- {eq}\frac{d}{dx}x^n=nx^{n-1} {/eq}

Simply put, if given a basic power function of the form {eq}x^n {/eq}, its derivative is given by bringing down the power of x (n) and multiplying it to the function, and then subtracting one from the exponent of x. If the power function has a coefficient, then its derivative will be:

- {eq}\frac{d}{dx}ax^n=nax^{n-1} {/eq}. This is also a form of the
**constant multiple rule**which states that {eq}\frac{d}{dx}(cf(x))=c*\frac{d}{dx}f(x)=cf'(x) {/eq}.

Here is the proof of the power rule for {eq}x^n {/eq}:

#### Proof:

- Start with the limit definition of the derivative: {eq}\frac{d}{dx}f(x)=\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h} {/eq}.

- Now, substitute the power function for f(x): {eq}\lim_{h \rightarrow 0}\frac{(x+h)^n-x^n}{h} {/eq}.

- Now, expand the {eq}(x+h)^n {/eq} term so the limit becomes: {eq}\lim_{h \rightarrow 0}\frac{(x+h)(x+h)...(x+h)-x^n}{h} {/eq}. In this limit, there are n copies of (x+h) being multiplied.

- In this product of n-terms, the leading term will be {eq}x^n {/eq} and the next term following that one will be {eq}nhx^{n-1} {/eq} followed by many terms all containing powers of h.

- Since the limit is subtracting {eq}x^n {/eq}, the limit then becomes: {eq}\lim_{h \rightarrow 0}\frac{nhx^{n-1}+...}{h} {/eq}. In this limit, the ellipsis indicates the terms following the initial term that all contain higher powers of h.

- Dividing all the terms by h gives: {eq}\lim_{h \rightarrow 0}nx^{n-1}+... {/eq}

- Finally, as {eq}h \rightarrow 0 {/eq}, the terms indicated by the ellipsis become zero which leaves the final term of {eq}nx^{n-1} {/eq}.

- Therefore, the derivative becomes: {eq}\frac{d}{dx}x^n=nx^{n-1} \square {/eq}

This shows that the power rule for the derivative of a power function will work for any power function. Next are some examples of the power rule.

### Power Rule Examples

Here are some examples of using the power rule to find the derivative of a power function (note that {eq}f'(x) {/eq} denotes the derivative of f(x).):

- Let {eq}f(x)=2x^2 {/eq}. Then {eq}f'(x)=(2)(2)x^{2-1}=4x^1=4x {/eq}.

- Let {eq}f(x)=-x^{3} {/eq}. Then {eq}f'(x)=(3)(-1)x^{3-1}=-3x^{2} {/eq}.

- Consider {eq}f(x)=1000x^1 {/eq}. Then {eq}f'(x)=(1)(1000)x^{1-1}=1000x^{0}=1000 {/eq}.

- Given {eq}f(x)=3x^7 {/eq} then its derivative would be {eq}f'(x)=(7)(3)x^{7-1}=21x^6 {/eq}.

- Let {eq}f(x)=6x^{5} {/eq}. Its derivative would then be {eq}f'(x)=(5)(6)x^{5-1}=30x^{4} {/eq}.

## Derivatives of Power Functions

In the preceding sections, only basic power functions were considered. But, what if instead of a single power function, there is a sum or difference of different power functions? This is actually called a polynomial and has the form {eq}f(x)=a_{n}x^n+a_{n-1}x^{n-1}+ ... + a_{1}x^{1}+a_{0} {/eq} where the {eq}a_{i} {/eq} are constant coefficients.

Now, using the **power rule**, **constant multiple rule**, and the **sum rule**, the derivatives of polynomial functions like the ones described above can be found. Moreover, the **sum rule states** that that states that: {eq}\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)=f'(x)+g'(x) {/eq}. This essentially says that the derivative of a sum is the sum of the derivatives.

Therefore, in order to differentiate polynomials of power functions, simply differentiate each term of the polynomial function independently and then add them together. Here is an example involving the general form of the polynomial:

- {eq}\frac{d}{dx}(f(x)=a_{n}x^n+a_{n-1}x^{n-1}+ ... + a_{1}x^{1}+a_{0})= {/eq}

- {eq}\frac{d}{dx}a_{n}x^n+\frac{d}{dx}a_{n-1}x^{n-1}+ ... + \frac{d}{dx}a_{1}x^{1}+\frac{d}{dx}a_{0}= {/eq}

- {eq}na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+ ... + (1)a_{1}x^{0}+(0)a_{0}= {/eq}

- {eq}na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+ ... + a_{1} {/eq}

Now, what is the exponent of x is negative? Is it still possible to differentiate the power function if it has negative exponents? The answer is **yes**. The method to differentiate power functions with negative powers is identical to the power rule formula used for power functions with positive exponents. Here is an explanation using {eq}x^{-n} {/eq}:

{eq}\frac{d}{dx}x^{-n}=(-n)x^{-n-1} {/eq}.

The power rule works exactly the same as in the positive exponent case. However, the value being brought down and multiplied to the power function is now negative and the exponent becomes more negative after differentiation is performed. Here are some examples to show these new properties:

- Let {eq}f(x)=x^3-2x^2+3x+4 {/eq}. Its derivative would then be: {eq}f'(x)=3x^{3-1}-(2)(2)x^{2-1}+(1)(3)x^{1-1}+(0)4 {/eq} which then becomes: {eq}f'(x)=3x^2-4x+3 {/eq}.

- Consider the power function {eq}f(x)=x^{-3}-x^{-1}+x^2 {/eq}. Its derivative would then be: {eq}f'(x)=(-3)x^{-3-1}-(-1)x^{-1-1}+(2)x^{2-1} {/eq} which then becomes: {eq}f'(x)=-3x^{-4}+x^{-2}+2x {/eq}.

- Let {eq}f(x)=x^5+x^{-4}+3x {/eq}. Then its derivative would be {eq}f'(x)=5x^{5-1}+(-4)x^{-4-1}+(1)(3)x^{1-1} {/eq} which when simplified becomes: {eq}f'(x)=5x^{4}-4x^{-5}+3 {/eq}.

- Consider the function {eq}f(x)=\frac{1}{x^2} {/eq}. It may not seem like a power function but it is. It is equal to: {eq}f(x)=\frac{1}{x^2}=x^{-2} {/eq} so its derivative would be: {eq}f'(x)=\frac{1}{x}+x^2+3 {/eq}.

- Consider the polynomial {eq}f(x)=3x^3-x^{-2}+7x^{-1} {/eq}. Using the rules above, its derivative is then: {eq}f'(x)=(3)(3)x^{3-1}-(-2)x^{-2-1}+(-1)(7)x^{-1-1} {/eq} which when simplified becomes: {eq}f'(x)=9x^2+2x^{-3}-7x^{-2} {/eq}.

## Lesson Summary

In this lesson, the power rule for calculating the derivative of a power function was derived and many examples were shown. Firstly, a **power function** is any function of the form {eq}f(x)=ax^n {/eq} where a and n are real numbers not equal to zero (otherwise the function would be trivial). The calculation of the derivative would be tedious but the power rule makes it simple. The power rule for the derivative of a power function is {eq}\frac{d}{dx}(ax^n)=nax^{n-1} {/eq}. Simply put, the derivative of a power function involves bringing the exponent value down and multiplying it to the function and then subtracting one from the exponent. The proof of this was also shown along with several examples.

Finally, the case of the sum of multiple power functions, which is called a polynomial, and the case of negative exponent were explored. The **sum rule for derivatives** and the constant multiple rule for derivatives were used in these calculations. In the polynomial case, the power rule applies to each of the individual power function terms. Therefore, in order find the derivative of a polynomial, simply differentiate each term individually using the power rules. Now, for power functions that have negative exponents, the power rule will also work. The only difference compared to the positive case is that the number brought down and multiplied is negative and the exponent becomes more negative. Here are the highlights:

- A
**power function**is a function of the form {eq}f(x)=ax^n {/eq} where a, n are both real numbers and both nonzero.

- The
**power rule**for the derivative of a power function is {eq}\frac{d}{dx}(ax^n)=nax^{n-1} {/eq}.

- The
**power rule**for the sum of power functions (polynomial) will work on the individual terms of the polynomial. Simply differentiate each term individually. - The constant multiple rule was also used throughout this lesson. It states that a constant multiplied to a function is then multiplied to the functions derivative: {eq}(cf(x))'=cf'(x) {/eq}.

- The power rule also works for power functions that have negative exponents. The power rule will instead bring down a negative number rather a positive number and the exponent becomes more negative since one is subtracted from the negative exponent.

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## Why Use the Power Rule When There Is Already a Definition?

How do we know the power rule is equivalent to the derivative given by using the limit definition of the derivative? Recall that the definition of the derivative of a function f(x) is given by

## Examples

Find the derivative of the following functions using the definition of the derivative and then by using the power rule to show that both results are equal.

## Solutions

1. For f(x), using the definition, we get

Using the power rule instead, we get

The answers agree and the power rule was much quicker!

2. For g(x), using the definition, we get

Using the power rule instead, we get

The answers agree again, and the power rule was much simpler to use.

3. For h(x), using the definition we get

Using the power rule instead, we get

The answers agree again and the power rule was, once again, much easier to use.

## Why Does This Work?

Although the examples above are not a proof, they likely convinced you that the power rule works. The actual proof is done the same way - using the limit definition of the derivative for the function x to the nth power. However, if you do not know the Binomial Theorem, it can be tricky to write out. The idea is that when you expand (x+h) to any power, you get a term with just x to a power, and then every single other term has an h in it. The second term in the expansion is always equal to nx^(n-1) h , and all the other terms with h's in them have h to a power higher than 1. The first term of the expansion, x^n, disappears when combining f(x+h) - f(x) in the numerator of our definition. Then, all other terms have an h cancel out with the h in the denominator - your first term of what is left is now nx^(n-1) and all other terms still have at least one h multiplied in them. So, when h goes to 0, all terms vanish except for the term given by the power rule: nx^(n-1). If that was hard to follow - try reading through it again while looking at the examples done above.

#### How does the power rule work?

The power rules works on power functions of the form f(x)=ax^n. The rule finds the derivative by first bringing the power of x (n) down and multiplying it to the function and then subtracting one from the power. The derivative is then f'(x)=nax^(n-1).

#### How do you find the derivative when the variable is in the exponent?

If the variable is in the exponent, the function has the form f(x)=a(b)^x with a,b real numbers. The power rule described in this lesson only works for power functions of the form f(x)=ax^n where a, n are real numbers and both nonzero. For the exponential function f(x)=a(b)^x, the derivative is given by f'(x)=a*ln(b)*(b)^x but it is not part of this lesson.

#### What is the power rule in exponents?

The power rule for the derivative of a power function is (ax^n)'=nax^(n-1). That is, if a function f(x)=ax^n is given with a, n both real numbers and nonzero, then its derivative is given by f'(x)=nax^(n-1) (bring down the power and multiply it to the function and then subtract one from the power).

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