Calculus Examples

$y = \frac{1}{3} x^{3} - 2 x + 7$

Step 1

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

$y = \frac{x^{3}}{3} - 2 x + 7$

Step 2

Set $y$ as a function of $x$ .

$f (x) = \frac{x^{3}}{3} - 2 x + 7$

Step 3

Find the derivative.

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Step 3.1

By the Sum Rule, the derivative of $\frac{x^{3}}{3} - 2 x + 7$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [7]$

Step 3.2

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

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Step 3.2.1

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

$\frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [7]$

Step 3.2.2

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

$\frac{1}{3} (3 x^{2}) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [7]$

Step 3.2.3

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

$\frac{3}{3} x^{2} + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [7]$

Step 3.2.4

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

$\frac{3 x^{2}}{3} + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [7]$

Step 3.2.5

Cancel the common factor of $3$ .

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Step 3.2.5.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [7]$

Step 3.2.5.2

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

$x^{2} + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [7]$

Step 3.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 3.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$x^{2} - 2 \frac{d}{d x} [x] + \frac{d}{d x} [7]$

Step 3.3.2

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

$x^{2} - 2 \cdot 1 + \frac{d}{d x} [7]$

Step 3.3.3

Multiply $- 2$ by $1$ .

$x^{2} - 2 + \frac{d}{d x} [7]$

Step 3.4

Differentiate using the Constant Rule.

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Step 3.4.1

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$x^{2} - 2 + 0$

Step 3.4.2

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

$x^{2} - 2$

Step 4

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

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Step 4.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 4.2

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

$x = \pm \sqrt{2}$

Step 4.3

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

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Step 4.3.1

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

$x = \sqrt{2}$

Step 4.3.2

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

$x = - \sqrt{2}$

Step 4.3.3

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

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

Step 5

Solve the original function $f (x) = \frac{x^{3}}{3} - 2 x + 7$ at $x = \sqrt{2}$ .

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Step 5.1

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

$f (\sqrt{2}) = \frac{{(\sqrt{2})}^{3}}{3} - 2 \sqrt{2} + 7$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

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

$f (\sqrt{2}) = \frac{\sqrt{2^{3}}}{3} - 2 \sqrt{2} + 7$

Step 5.2.1.2

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

$f (\sqrt{2}) = \frac{\sqrt{8}}{3} - 2 \sqrt{2} + 7$

Step 5.2.1.3

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

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Step 5.2.1.3.1

Factor $4$ out of $8$ .

$f (\sqrt{2}) = \frac{\sqrt{4 (2)}}{3} - 2 \sqrt{2} + 7$

Step 5.2.1.3.2

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

$f (\sqrt{2}) = \frac{\sqrt{2^{2} \cdot 2}}{3} - 2 \sqrt{2} + 7$

Step 5.2.1.4

Pull terms out from under the radical.

$f (\sqrt{2}) = \frac{2 \sqrt{2}}{3} - 2 \sqrt{2} + 7$

Step 5.2.2

To write $- 2 \sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\sqrt{2}) = \frac{2 \sqrt{2}}{3} - 2 \sqrt{2} \cdot \frac{3}{3} + 7$

Step 5.2.3

Combine fractions.

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Step 5.2.3.1

Combine $- 2 \sqrt{2}$ and $\frac{3}{3}$ .

$f (\sqrt{2}) = \frac{2 \sqrt{2}}{3} + \frac{- 2 \sqrt{2} \cdot 3}{3} + 7$

Step 5.2.3.2

Combine the numerators over the common denominator.

$f (\sqrt{2}) = \frac{2 \sqrt{2} - 2 \sqrt{2} \cdot 3}{3} + 7$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Simplify the numerator.

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Step 5.2.4.1.1

Multiply $3$ by $- 2$ .

$f (\sqrt{2}) = \frac{2 \sqrt{2} - 6 \sqrt{2}}{3} + 7$

Step 5.2.4.1.2

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

$f (\sqrt{2}) = \frac{- 4 \sqrt{2}}{3} + 7$

Step 5.2.4.2

Move the negative in front of the fraction.

$f (\sqrt{2}) = - \frac{4 \sqrt{2}}{3} + 7$

Step 5.2.5

The final answer is $- \frac{4 \sqrt{2}}{3} + 7$ .

$- \frac{4 \sqrt{2}}{3} + 7$

Step 6

Solve the original function $f (x) = \frac{x^{3}}{3} - 2 x + 7$ at $x = - \sqrt{2}$ .

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Step 6.1

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

$f (- \sqrt{2}) = \frac{{(- \sqrt{2})}^{3}}{3} - 2 (- \sqrt{2}) + 7$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

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

$f (- \sqrt{2}) = \frac{{(- 1)}^{3} {\sqrt{2}}^{3}}{3} - 2 (- \sqrt{2}) + 7$

Step 6.2.1.1.2

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

$f (- \sqrt{2}) = \frac{- {\sqrt{2}}^{3}}{3} - 2 (- \sqrt{2}) + 7$

Step 6.2.1.1.3

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

$f (- \sqrt{2}) = \frac{- \sqrt{2^{3}}}{3} - 2 (- \sqrt{2}) + 7$

Step 6.2.1.1.4

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

$f (- \sqrt{2}) = \frac{- \sqrt{8}}{3} - 2 (- \sqrt{2}) + 7$

Step 6.2.1.1.5

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

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Step 6.2.1.1.5.1

Factor $4$ out of $8$ .

$f (- \sqrt{2}) = \frac{- \sqrt{4 (2)}}{3} - 2 (- \sqrt{2}) + 7$

Step 6.2.1.1.5.2

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

$f (- \sqrt{2}) = \frac{- \sqrt{2^{2} \cdot 2}}{3} - 2 (- \sqrt{2}) + 7$

Step 6.2.1.1.6

Pull terms out from under the radical.

$f (- \sqrt{2}) = \frac{- 1 \cdot (2 \sqrt{2})}{3} - 2 (- \sqrt{2}) + 7$

Step 6.2.1.1.7

Multiply $- 1$ by $2$ .

$f (- \sqrt{2}) = \frac{- 2 \sqrt{2}}{3} - 2 (- \sqrt{2}) + 7$

Step 6.2.1.2

Move the negative in front of the fraction.

$f (- \sqrt{2}) = - \frac{2 \sqrt{2}}{3} - 2 (- \sqrt{2}) + 7$

Step 6.2.1.3

Multiply $- 1$ by $- 2$ .

$f (- \sqrt{2}) = - \frac{2 \sqrt{2}}{3} + 2 \sqrt{2} + 7$

Step 6.2.2

To write $2 \sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \sqrt{2}) = - \frac{2 \sqrt{2}}{3} + 2 \sqrt{2} \cdot \frac{3}{3} + 7$

Step 6.2.3

Combine fractions.

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Step 6.2.3.1

Combine $2 \sqrt{2}$ and $\frac{3}{3}$ .

$f (- \sqrt{2}) = - \frac{2 \sqrt{2}}{3} + \frac{2 \sqrt{2} \cdot 3}{3} + 7$

Step 6.2.3.2

Combine the numerators over the common denominator.

$f (- \sqrt{2}) = \frac{- 2 \sqrt{2} + 2 \sqrt{2} \cdot 3}{3} + 7$

Step 6.2.4

Simplify the numerator.

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Step 6.2.4.1

Multiply $3$ by $2$ .

$f (- \sqrt{2}) = \frac{- 2 \sqrt{2} + 6 \sqrt{2}}{3} + 7$

Step 6.2.4.2

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

$f (- \sqrt{2}) = \frac{4 \sqrt{2}}{3} + 7$

Step 6.2.5

The final answer is $\frac{4 \sqrt{2}}{3} + 7$ .

$\frac{4 \sqrt{2}}{3} + 7$

Step 7

The horizontal tangent lines on function $f (x) = \frac{x^{3}}{3} - 2 x + 7$ are $y = - \frac{4 \sqrt{2}}{3} + 7, y = \frac{4 \sqrt{2}}{3} + 7$ .

$y = - \frac{4 \sqrt{2}}{3} + 7, y = \frac{4 \sqrt{2}}{3} + 7$

Step 8