Calculus Examples

$f (x) = x^{3} - 6 x$

Step 1

Find the derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{3} - 6 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x]$

Step 1.1.2

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

$3 x^{2} + \frac{d}{d x} [- 6 x]$

Step 1.2

Evaluate $\frac{d}{d x} [- 6 x]$ .

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Step 1.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$3 x^{2} - 6 \frac{d}{d x} [x]$

Step 1.2.2

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

$3 x^{2} - 6 \cdot 1$

Step 1.2.3

Multiply $- 6$ by $1$ .

$3 x^{2} - 6$

Step 2

Set the derivative equal to $0$ then solve the equation $3 x^{2} - 6 = 0$ .

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Step 2.1

Add $6$ to both sides of the equation.

$3 x^{2} = 6$

Step 2.2

Divide each term in $3 x^{2} = 6$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x^{2} = 6$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{6}{3}$

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.

$\frac{3 x^{2}}{3} = \frac{6}{3}$

Step 2.2.2.1.2

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

$x^{2} = \frac{6}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $6$ by $3$ .

$x^{2} = 2$

Step 2.3

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

$x = \pm \sqrt{2}$

Step 2.4

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

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Step 2.4.1

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

$x = \sqrt{2}$

Step 2.4.2

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

$x = - \sqrt{2}$

Step 2.4.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 3

Solve the original function $f (x) = x^{3} - 6 x$ at $x = \sqrt{2}$ .

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Step 3.1

Replace the variable $x$ with $\sqrt{2}$ in the expression.

$f (\sqrt{2}) = {(\sqrt{2})}^{3} - 6 \sqrt{2}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

$f (\sqrt{2}) = \sqrt{2^{3}} - 6 \sqrt{2}$

Step 3.2.1.2

Raise $2$ to the power of $3$ .

$f (\sqrt{2}) = \sqrt{8} - 6 \sqrt{2}$

Step 3.2.1.3

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.2.1.3.1

Factor $4$ out of $8$ .

$f (\sqrt{2}) = \sqrt{4 (2)} - 6 \sqrt{2}$

Step 3.2.1.3.2

Rewrite $4$ as $2^{2}$ .

$f (\sqrt{2}) = \sqrt{2^{2} \cdot 2} - 6 \sqrt{2}$

Step 3.2.1.4

Pull terms out from under the radical.

$f (\sqrt{2}) = 2 \sqrt{2} - 6 \sqrt{2}$

Step 3.2.2

Subtract $6 \sqrt{2}$ from $2 \sqrt{2}$ .

$f (\sqrt{2}) = - 4 \sqrt{2}$

Step 3.2.3

The final answer is $- 4 \sqrt{2}$ .

$- 4 \sqrt{2}$

Step 4

Solve the original function $f (x) = x^{3} - 6 x$ at $x = - \sqrt{2}$ .

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Step 4.1

Replace the variable $x$ with $- \sqrt{2}$ in the expression.

$f (- \sqrt{2}) = {(- \sqrt{2})}^{3} - 6 (- \sqrt{2})$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Apply the product rule to $- \sqrt{2}$ .

$f (- \sqrt{2}) = {(- 1)}^{3} {\sqrt{2}}^{3} - 6 (- \sqrt{2})$

Step 4.2.1.2

Raise $- 1$ to the power of $3$ .

$f (- \sqrt{2}) = - {\sqrt{2}}^{3} - 6 (- \sqrt{2})$

Step 4.2.1.3

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

$f (- \sqrt{2}) = - \sqrt{2^{3}} - 6 (- \sqrt{2})$

Step 4.2.1.4

Raise $2$ to the power of $3$ .

$f (- \sqrt{2}) = - \sqrt{8} - 6 (- \sqrt{2})$

Step 4.2.1.5

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.2.1.5.1

Factor $4$ out of $8$ .

$f (- \sqrt{2}) = - \sqrt{4 (2)} - 6 (- \sqrt{2})$

Step 4.2.1.5.2

Rewrite $4$ as $2^{2}$ .

$f (- \sqrt{2}) = - \sqrt{2^{2} \cdot 2} - 6 (- \sqrt{2})$

Step 4.2.1.6

Pull terms out from under the radical.

$f (- \sqrt{2}) = - (2 \sqrt{2}) - 6 (- \sqrt{2})$

Step 4.2.1.7

Multiply $2$ by $- 1$ .

$f (- \sqrt{2}) = - 2 \sqrt{2} - 6 (- \sqrt{2})$

Step 4.2.1.8

Multiply $- 1$ by $- 6$ .

$f (- \sqrt{2}) = - 2 \sqrt{2} + 6 \sqrt{2}$

Step 4.2.2

Add $- 2 \sqrt{2}$ and $6 \sqrt{2}$ .

$f (- \sqrt{2}) = 4 \sqrt{2}$

Step 4.2.3

The final answer is $4 \sqrt{2}$ .

$4 \sqrt{2}$

Step 5

The horizontal tangent lines on function $f (x) = x^{3} - 6 x$ are $y = - 4 \sqrt{2}, y = 4 \sqrt{2}$ .

$y = - 4 \sqrt{2}, y = 4 \sqrt{2}$

Step 6