Calculus Examples

$x^{3} + x$

Step 1

Find the derivative.

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Step 1.1

By the Sum Rule, the derivative of $x^{3} + x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [x]$

Step 1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 x^{2} + 1$

Step 2

Set the derivative equal to $0$ then solve the equation $3 x^{2} + 1 = 0$ .

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Step 2.1

Subtract $1$ from both sides of the equation.

$3 x^{2} = - 1$

Step 2.2

Divide each term in $3 x^{2} = - 1$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x^{2} = - 1$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{- 1}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- 1}{3}$

Step 2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 1}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{3}$

Step 2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- \frac{1}{3}}$

Step 2.4

Simplify $\pm \sqrt{- \frac{1}{3}}$ .

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Step 2.4.1

Rewrite $- \frac{1}{3}$ as $i^{2} \frac{1^{2}}{3}$ .

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Step 2.4.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1}{3}}$

Step 2.4.1.2

Rewrite $1$ as $1^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1^{2}}{3}}$

Step 2.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{3}}$

Step 2.4.3

One to any power is one.

$x = \pm i \sqrt{\frac{1}{3}}$

Step 2.4.4

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{3}}$

Step 2.4.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{3}}$

Step 2.4.6

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 2.4.7

Combine and simplify the denominator.

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Step 2.4.7.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.4.7.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.4.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.4.7.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.4.7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.4.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm i \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.4.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm i \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.4.7.6.4

Cancel the common factor of $2$ .

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Step 2.4.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.4.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{3}}{3^{1}}$

Step 2.4.7.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{3}}{3}$

Step 2.4.8

Combine $i$ and $\frac{\sqrt{3}}{3}$ .

$x = \pm \frac{i \sqrt{3}}{3}$

Step 2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{i \sqrt{3}}{3}$

Step 2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{i \sqrt{3}}{3}$

Step 2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{i \sqrt{3}}{3}, - \frac{i \sqrt{3}}{3}$

Step 3

A tangent line cannot be found at an imaginary point. The point at $x =$ does not exist on the real coordinate system.

A tangent cannot be found from the root

Step 4

There are no horizontal tangent lines on the function $f (x) = x^{3} + x$ .

No horizontal tangent lines

Step 5