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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.

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.

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.

Okay, now you are ready to try your hand at working some basic practice problems. Unless otherwise instructed, calculate the derivatives of the following functions using the power rule and the basic rules on the derivatives page (but do not use the limit definition of the derivative) giving your answers in simplified form.

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. Here is a list converting the old numbering system to the new.

Derivative Power Rule- Practice Problems Conversion

[A01-897] - [A02-909] - [A03-910] - [A04-911] - [A05-912] - [A06-913] - [A07-914] - [A08-915] - [A09-916]

[A10-917] - [A11-918] - [B01-919] - [B02-920] - [B03-921] - [B04-922] - [B05-923] - [B06-924]

Please update your notes to this new numbering system. The display of this conversion information is temporary.

GOT IT. THANKS!

\( f(x) = x^{10} \)

Problem Statement

Calculate the derivative of \( f(x) = x^{10} \).

Final Answer

\( 10x^9 \)

Problem Statement

Calculate the derivative of \( f(x) = x^{10} \).

Solution

917 solution video

video by PatrickJMT

Final Answer

\( 10x^9 \)

close solution

\( y = 3x^2 \)

Problem Statement

Calculate the derivative of \( y = 3x^2 \).

Final Answer

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

Problem Statement

Calculate the derivative of \( y = 3x^2 \).

Solution

\(\displaystyle{\frac{d}{dx}\left[3x^2\right]=3\frac{d[x^2]}{dx}=3(2x^1)=6x}\)

Final Answer

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

close solution

\( y = 7x^3 \)

Problem Statement

Calculate the derivative of \( y = 7x^3 \).

Final Answer

\(\displaystyle{\frac{d}{dx}[7x^3]=21x^2}\)

Problem Statement

Calculate the derivative of \( y = 7x^3 \).

Solution

\(\displaystyle{\frac{d}{dx}[7x^3]=7\frac{d[x^3]}{dx}=7(3x^2)=21x^2}\)

Final Answer

\(\displaystyle{\frac{d}{dx}[7x^3]=21x^2}\)

close solution

\( y = 5x^4 \)

Problem Statement

Calculate the derivative of \( y = 5x^4 \).

Final Answer

\(\displaystyle{\frac{d}{dx}\left[5x^4\right]=20x^3}\)

Problem Statement

Calculate the derivative of \( y = 5x^4 \).

Solution

\(\displaystyle{\frac{d}{dx}\left[5x^4\right]=5\frac{d[x^4]}{dx}=5(4x^3)=20x^3}\)

Final Answer

\(\displaystyle{\frac{d}{dx}\left[5x^4\right]=20x^3}\)

close solution

\( g(y) = y^4 \)

Problem Statement

Calculate the derivative of \( g(y) = y^4 \).

Final Answer

\(\displaystyle{\frac{d}{dy}\left[y^4\right]=4y^3}\)

Problem Statement

Calculate the derivative of \( g(y) = y^4 \).

Solution

\(\displaystyle{\frac{d}{dy}\left[y^4\right]=4y^{4-1}=4y^3}\)

Final Answer

\(\displaystyle{\frac{d}{dy}\left[y^4\right]=4y^3}\)

close solution

\( y = 2x^2 + 3x - 17 \)

Problem Statement

Calculate the derivative of \( y = 2x^2 + 3x - 17 \).

Final Answer

\( y' = 4x + 3\)

Problem Statement

Calculate the derivative of \( y = 2x^2 + 3x - 17 \).

Solution

909 solution video

video by Krista King Math

Final Answer

\( y' = 4x + 3\)

close solution

\( f(x) = 100 - 16x^2 \)

Problem Statement

Calculate the derivative of \( f(x) = 100 - 16x^2 \).

Final Answer

\( f'(x) = -32x \)

Problem Statement

Calculate the derivative of \( f(x) = 100 - 16x^2 \).

Solution

910 solution video

video by Krista King Math

Final Answer

\( f'(x) = -32x \)

close solution

\( f(x) = 4x - 5 \)

Problem Statement

Calculate the derivative of \( f(x) = 4x - 5 \).

Final Answer

\( f'(x) = 4 \)

Problem Statement

Calculate the derivative of \( f(x) = 4x - 5 \).

Solution

911 solution video

video by Krista King Math

Final Answer

\( f'(x) = 4 \)

close solution

\( y = x^6 - 7x^4 + 5/x \)

Problem Statement

Calculate the derivative of \( y = x^6 - 7x^4 + 5/x \).

Final Answer

\( y' = 6x^5 - 28x^3 - 5x^{-2} \)

Problem Statement

Calculate the derivative of \( y = x^6 - 7x^4 + 5/x \).

Solution

912 solution video

video by Krista King Math

Final Answer

\( y' = 6x^5 - 28x^3 - 5x^{-2} \)

close solution

\( g(x) = x^{5/2} + 3x + 4 \)

Problem Statement

Calculate the derivative of \( g(x) = x^{5/2} + 3x + 4 \).

Solution

918 solution video

video by PatrickJMT

close solution

\(5x^4-6x+90x^{100}\)

Problem Statement

\(5x^4-6x+90x^{100}\)

Solution

2303 solution video

video by Krista King Math

close solution

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 [9min-6secs]

video by PatrickJMT

Okay, try some of these intermediate level practice problems. Just as before, do not use the limit definition to calculate the derivative and give your answers in simplified form.

\( f(x) = 2x^4 + 3x^{5/3} - 4/x \)

Problem Statement

Calculate the derivative of \( f(x) = 2x^4 + 3x^{5/3} - 4/x \).

Solution

921 solution video

video by PatrickJMT

close solution

\(\displaystyle{ h(x) = \frac{1}{\sqrt[3]{x}} }\)

Problem Statement

Calculate the derivative of \(\displaystyle{ h(x) = \frac{1}{\sqrt[3]{x}} }\).

Solution

919 solution video

video by PatrickJMT

close solution

\(\displaystyle{ p(q) = \frac{-3}{\pi}\sqrt{q^{4.1}} }\)

Problem Statement

Calculate the derivative of \(\displaystyle{ p(q) = \frac{-3}{\pi}\sqrt{q^{4.1}} }\).

Solution

920 solution video

video by PatrickJMT

close solution

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

Problem Statement

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

Solution

922 solution video

video by PatrickJMT

close solution

\( y = \sqrt[5]{x} + 4\sqrt{x^5} \)

Problem Statement

Calculate the derivative of \( y = \sqrt[5]{x} + 4\sqrt{x^5} \).

Solution

923 solution video

video by Krista King Math

close solution

\(\displaystyle{y=\frac{x^2+4x+3}{\sqrt{x}}}\)

Problem Statement

Calculate the derivative of \(\displaystyle{y=\frac{x^2+4x+3}{\sqrt{x}}}\).

Solution

924 solution video

video by Krista King Math

close solution

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, let's work some practice problems calculating higher order derivatives. As before, do not use the limit definition to calculate the derivative and give your answers in completely factored simplified form.

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

Problem Statement

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

Final Answer

\(\displaystyle{6(6x^2+1)}\)

Problem Statement

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

Solution

The first derivative is \(\displaystyle{\frac{d}{dx}[3x^4+3x^2+2x]=12x^3+6x+2}\)
To find the second derivative, we take the derivative of the first derivative.
\(\displaystyle{\frac{d}{dx}[12x^3+6x+2]=36x^2+6}\)

Final Answer

\(\displaystyle{6(6x^2+1)}\)

close solution

Okay, you are ready to move on. If you need a suggestion on where to go next,
the product rule is the logical next step.

 
product rule →
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