## 17Calculus Derivatives - Power Rule

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:

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.

Basic Practice

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.

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$f(x) = x^{10}$$

Final Answer

$$10x^9$$

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$f(x) = x^{10}$$

Solution

### 917 video

video by PatrickJMT

Final Answer

$$10x^9$$

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$$y = 3x^2$$

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$y = 3x^2$$

Final Answer

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$y = 3x^2$$

Solution

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

Final Answer

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

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$$y = 7x^3$$

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$y = 7x^3$$

Final Answer

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$y = 7x^3$$

Solution

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

Final Answer

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

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$$y = 5x^4$$

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$y = 5x^4$$

Final Answer

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$y = 5x^4$$

Solution

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

Final Answer

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

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$$g(y) = y^4$$

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$g(y) = y^4$$

Final Answer

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$g(y) = y^4$$

Solution

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

Final Answer

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

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$$y = 2x^2 + 3x - 17$$

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$y = 2x^2 + 3x - 17$$

Final Answer

$$y' = 4x + 3$$

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$y = 2x^2 + 3x - 17$$

Solution

### 909 video

video by Krista King Math

Final Answer

$$y' = 4x + 3$$

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$f(x) = 100 - 16x^2$$

Final Answer

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$f(x) = 100 - 16x^2$$

Solution

### 910 video

video by Krista King Math

Final Answer

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

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$f(x) = 4x - 5$$

Final Answer

$$f'(x) = 4$$

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$f(x) = 4x - 5$$

Solution

### 911 video

video by Krista King Math

Final Answer

$$f'(x) = 4$$

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$$y = x^6 - 7x^4 + 5/x$$

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$y = x^6 - 7x^4 + 5/x$$

Final Answer

$$y' = 6x^5 - 28x^3 - 5x^{-2}$$

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$y = x^6 - 7x^4 + 5/x$$

Solution

### 912 video

video by Krista King Math

Final Answer

$$y' = 6x^5 - 28x^3 - 5x^{-2}$$

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$g(x) = x^{5/2} + 3x + 4$$

Solution

### 918 video

video by PatrickJMT

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$$5x^4-6x+90x^{100}$$

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$5x^4-6x+90x^{100}$$

Solution

### 2303 video

video by Krista King Math

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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\$$ in \$$y^3\$$ 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.

Intermediate Practice

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.

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$f(x) = 2x^4 + 3x^{5/3} - 4/x$$

Solution

### 921 video

video by PatrickJMT

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$\displaystyle{ h(x) = \frac{1}{\sqrt[3]{x}} }$$

Solution

### 919 video

video by PatrickJMT

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$\displaystyle{ p(q) = \frac{-3}{\pi}\sqrt{q^{4.1}} }$$

Solution

### 920 video

video by PatrickJMT

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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 video

video by PatrickJMT

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$y = \sqrt[5]{x} + 4\sqrt{x^5}$$

Solution

### 923 video

video by Krista King Math

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

Problem Statement

Calculate the derivative of this function using the power rule and the basic derivative rules (but do not use the limit definition of the derivative). Give your answer in exact, simplified form. $$\displaystyle{y=\frac{x^2+4x+3}{\sqrt{x}}}$$

Solution

### 924 video

video by Krista King Math

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 Okay, you are ready to move on. If you need a suggestion on where to go next, calculating higher order derivatives is the logical next step.

You CAN Ace Calculus

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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