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17calculus > derivatives > power rule

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Power Rule

on this page: ► how to use the power rule     ► when you can't use the power rule     ► higher order derivatives

Power Rule Theorem

For any rational number r, the function \(g(x)=x^r\) is differentiable, and \(\displaystyle{ \frac{d}{dx}[g(x)] = rx^{r-1} }\).

[ Note: We use the letter r here to emphasize that the exponent must be a rational number, i.e. it cannot be irrational, like \(\pi\), or a variable or anything other than a rational number. This detail will be extremely important very soon in your study of derivatives. ]

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How To Use The Power Rule

Basically the Power Rule Theorem says, if you have a term of the form \(x^r\) where r is a rational number, then the derivative of this is \(rx^{r-1}\). You think of this as taking the exponent and putting it in front of the term and then subtracting one from the exponent like this:
power rule
It helps to remember if you visualize the exponent moving, which is what we showed with the arrow. This idea works fine as long as there is no number in front of the x. If there is a number in front of the x, you just multiply the two numbers using the constant multiple rule. It looks like this: \( \displaystyle{ \frac{d}{dx}[ax^n] = a(nx^{n-1}) }\).
The reason it works like this is because we use the constant multiple rule first, then the power rule. It looks like this.

\(\displaystyle{ \frac{d}{dx}[ax^n] }\)


\(\displaystyle{ a\frac{d}{dx}[x^n] }\)

Constant Multiple Rule


\(\displaystyle{ a(nx^{n-1}) }\)

Power Rule

Constant Multiple Rule

\(\displaystyle{ \frac{d}{dx}[ax^n] = }\) \(\displaystyle{ a\frac{d}{dx}[x^n] }\)

Power Rule

\(\displaystyle{ a\frac{d}{dx}[x^n] }\) \(\displaystyle{ a(nx^{n-1}) }\)

Okay, let's do some examples.

Example 1:

\(\displaystyle{\frac{d}{dx}[x^3] = }\) \(\displaystyle{ 3x^{3-1} = 3x^2}\)

Example 2:

\(\displaystyle{\frac{d}{dx}[2x^3] = }\) \(\displaystyle{ 2\frac{d}{dx}[x^3] = }\) \(\displaystyle{ 2(3x^2) = 6x^2}\)

As you get more practice, you probably won't need to do the two middle steps as shown in Example 2. You will probably just think of taking the coefficient 2 times the exponent 3 to get 6 out front.

When You Can't Use The Power Rule

There are two main situations where you can not use the power rule when you have an exponent.
First, when the term that has a power does not exactly match the derivative variable. In this case, the chain rule is required. Here are two examples.

Example 3: \(\displaystyle{\frac{d}{dx}[y^3]}\)
Notice here that the variable 'y' does not match the 'x' in 'dx'. So you can't use the power rule here. You need to use the chain rule.

Example 4: \(\displaystyle{\frac{d}{dx}\left[ (x^2+5)^3\right]}\)
In this case, the term \( (x^2+5) \) does not exactly match the x in dx. So you can't use the power rule here either ( on the \(3\) power ). The chain rule is required.

The second main situation is when the exponent is not a rational number.
This is an important detail to remember (and an example of why you need to read theorems carefully). For example, you can't use the power rule on \( g(t) = 2^t \) because t is not a rational number. Notice the difference between \( 2^t \) [cannot use power rule] and \( t^2 \) [can use power rule].
Also, you cannot use the power rule on \( x^{\pi} \) since \( \pi \) is not rational.
In these cases logarithmic differentiation is the way to go.

Okay, before working practice problems, take a few minutes to just sit back and watch this video. It shows lots of examples using the power rule and the basic rules (constant rule, constant multiple rule and addition and subtraction rules).

PatrickJMT - Basic Derivative Examples

Now that you know the power rule and the basic rules on the main derivatives page, you will not use the limit definition to calculate derivatives, unless otherwise instructed to do so.

Higher Order Derivatives

Although higher order derivatives are not directly related to the power rule, you need to know the power rule in order to be able to understand examples of how to find higher order derivatives. So we include the discussion here.

Higher order derivatives are not very difficult. The idea is that, after taking the first derivative, you can take the derivative again to get the second derivative, and so on. Here is a quick example.
Example - - Find the second derivative of \(f(x)=3x^5 + 2x+1\).
The first derivative is \( f'(x) = 15x^4 + 2\).
Taking the derivative again, gives us the second derivative \( f''(x) = 60x^3 \).

Notation - - We can continue taking as many derivatives as we want. The third derivative is written \( f'''(x) \) or \( f^{(3)}(x)\). Notice that the derivative number is written in parentheses so that we can tell we are talking about a derivative and not a power, which we would write \( f^{3}(x)\). Starting with the fourth derivative, we abandon the 'multiple quote' notation and write \(f^{(4)}(x) \) and so on.

Okay, your next task is to work some practice problems.
After that, if you need a suggestion on where to go next, the product rule is the logical next step.

product rule →

Practice Problems

Instructions - - Unless otherwise instructed, calculate the derivatives of the following functions using the power rule and the basic rules on the derivatives page (but not the limit definition of the derivative) giving your answers in simplified form.

Level A - Basic

Practice A01

calculate \(f''(x)\) of \(f(x)=3x^4+3x^2+2x \)



Practice A02



Practice A03



Practice A04



Practice A05



Practice A06




Practice A07




Practice A08




Practice A09




Practice A10



Practice A11



Level B - Intermediate

Practice B01



Practice B02



Practice B03



Practice B04

What is \(f'(64)\) for \(\displaystyle{f(x)=x+3\sqrt{x}+4\sqrt[3]{x}}\)?


Practice B05



Practice B06



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