Calculus Examples

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

Step 1

Find the derivative.

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Step 1.1

By the Sum Rule, the derivative of $\frac{1}{3} x^{3} - 3 x + 2$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [2]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{1}{3} x^{3}]$ .

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Step 1.2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} x^{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [2]$

Step 1.2.2

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

$\frac{1}{3} (3 x^{2}) + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [2]$

Step 1.2.3

Combine $3$ and $\frac{1}{3}$ .

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

Step 1.2.4

Combine $\frac{3}{3}$ and $x^{2}$ .

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

Step 1.2.5

Cancel the common factor of $3$ .

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Step 1.2.5.1

Cancel the common factor.

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

Step 1.2.5.2

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

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

Step 1.3

Evaluate $\frac{d}{d x} [- 3 x]$ .

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Step 1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

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

Step 1.3.2

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

$x^{2} - 3 \cdot 1 + \frac{d}{d x} [2]$

Step 1.3.3

Multiply $- 3$ by $1$ .

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

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$x^{2} - 3 + 0$

Step 1.4.2

Add $x^{2} - 3$ and $0$ .

$x^{2} - 3$

Step 2

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

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Step 2.1

Add $3$ to both sides of the equation.

$x^{2} = 3$

Step 2.2

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

$x = \pm \sqrt{3}$

Step 2.3

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

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Step 2.3.1

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

$x = \sqrt{3}$

Step 2.3.2

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

$x = - \sqrt{3}$

Step 2.3.3

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

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

Step 3

Solve the original function $f (x) = \frac{1}{3} x^{3} - 3 x + 2$ at $x = \sqrt{3}$ .

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Step 3.1

Replace the variable $x$ with $\sqrt{3}$ in the expression.

$f (\sqrt{3}) = \frac{1}{3} \cdot {(\sqrt{3})}^{3} - 3 \sqrt{3} + 2$

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

$f (\sqrt{3}) = \frac{1}{3} \cdot \sqrt{3^{3}} - 3 \sqrt{3} + 2$

Step 3.2.1.2

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

$f (\sqrt{3}) = \frac{1}{3} \cdot \sqrt{27} - 3 \sqrt{3} + 2$

Step 3.2.1.3

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.2.1.3.1

Factor $9$ out of $27$ .

$f (\sqrt{3}) = \frac{1}{3} \cdot \sqrt{9 (3)} - 3 \sqrt{3} + 2$

Step 3.2.1.3.2

Rewrite $9$ as $3^{2}$ .

$f (\sqrt{3}) = \frac{1}{3} \cdot \sqrt{3^{2} \cdot 3} - 3 \sqrt{3} + 2$

Step 3.2.1.4

Pull terms out from under the radical.

$f (\sqrt{3}) = \frac{1}{3} \cdot (3 \sqrt{3}) - 3 \sqrt{3} + 2$

Step 3.2.1.5

Cancel the common factor of $3$ .

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Step 3.2.1.5.1

Factor $3$ out of $3 \sqrt{3}$ .

$f (\sqrt{3}) = \frac{1}{3} \cdot (3 (\sqrt{3})) - 3 \sqrt{3} + 2$

Step 3.2.1.5.2

Cancel the common factor.

$f (\sqrt{3}) = \frac{1}{3} \cdot (3 \sqrt{3}) - 3 \sqrt{3} + 2$

Step 3.2.1.5.3

Rewrite the expression.

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

Step 3.2.2

Subtract $3 \sqrt{3}$ from $\sqrt{3}$ .

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

Step 3.2.3

The final answer is $- 2 \sqrt{3} + 2$ .

$- 2 \sqrt{3} + 2$

Step 4

Solve the original function $f (x) = \frac{1}{3} x^{3} - 3 x + 2$ at $x = - \sqrt{3}$ .

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Step 4.1

Replace the variable $x$ with $- \sqrt{3}$ in the expression.

$f (- \sqrt{3}) = \frac{1}{3} \cdot {(- \sqrt{3})}^{3} - 3 (- \sqrt{3}) + 2$

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 $- \sqrt{3}$ .

$f (- \sqrt{3}) = \frac{1}{3} \cdot ({(- 1)}^{3} {\sqrt{3}}^{3}) - 3 (- \sqrt{3}) + 2$

Step 4.2.1.2

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

$f (- \sqrt{3}) = \frac{1}{3} \cdot (- {\sqrt{3}}^{3}) - 3 (- \sqrt{3}) + 2$

Step 4.2.1.3

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

$f (- \sqrt{3}) = \frac{1}{3} \cdot (- \sqrt{3^{3}}) - 3 (- \sqrt{3}) + 2$

Step 4.2.1.4

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

$f (- \sqrt{3}) = \frac{1}{3} \cdot (- \sqrt{27}) - 3 (- \sqrt{3}) + 2$

Step 4.2.1.5

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.2.1.5.1

Factor $9$ out of $27$ .

$f (- \sqrt{3}) = \frac{1}{3} \cdot (- \sqrt{9 (3)}) - 3 (- \sqrt{3}) + 2$

Step 4.2.1.5.2

Rewrite $9$ as $3^{2}$ .

$f (- \sqrt{3}) = \frac{1}{3} \cdot (- \sqrt{3^{2} \cdot 3}) - 3 (- \sqrt{3}) + 2$

Step 4.2.1.6

Pull terms out from under the radical.

$f (- \sqrt{3}) = \frac{1}{3} \cdot (- (3 \sqrt{3})) - 3 (- \sqrt{3}) + 2$

Step 4.2.1.7

Cancel the common factor of $3$ .

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Step 4.2.1.7.1

Factor $3$ out of $- (3 \sqrt{3})$ .

$f (- \sqrt{3}) = \frac{1}{3} \cdot (3 (- (\sqrt{3}))) - 3 (- \sqrt{3}) + 2$

Step 4.2.1.7.2

Cancel the common factor.

$f (- \sqrt{3}) = \frac{1}{3} \cdot (3 (- (\sqrt{3}))) - 3 (- \sqrt{3}) + 2$

Step 4.2.1.7.3

Rewrite the expression.

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

Step 4.2.1.8

Multiply $- 1$ by $- 3$ .

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

Step 4.2.2

Add $- \sqrt{3}$ and $3 \sqrt{3}$ .

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

Step 4.2.3

The final answer is $2 \sqrt{3} + 2$ .

$2 \sqrt{3} + 2$

Step 5

The horizontal tangent lines on function $f (x) = \frac{1}{3} x^{3} - 3 x + 2$ are $y = - 2 \sqrt{3} + 2, y = 2 \sqrt{3} + 2$ .

$y = - 2 \sqrt{3} + 2, y = 2 \sqrt{3} + 2$

Step 6