Calculus Examples

$f (x) = x^{3} - 4 x$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{3} - 4 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x]$ .

$f' (x) = \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 4 x)$

Step 1.1.1.2

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

$f' (x) = 3 x^{2} + \frac{d}{d x} (- 4 x)$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 4 x]$ .

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Step 1.1.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$f' (x) = 3 x^{2} - 4 \frac{d}{d x} x$

Step 1.1.2.2

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

$f' (x) = 3 x^{2} - 4 \cdot 1$

Step 1.1.2.3

Multiply $- 4$ by $1$ .

$f' (x) = 3 x^{2} - 4$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 4$ .

$3 x^{2} - 4$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} - 4 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$3 x^{2} - 4 = 0$

Step 2.2

Add $4$ to both sides of the equation.

$3 x^{2} = 4$

Step 2.3

Divide each term in $3 x^{2} = 4$ by $3$ and simplify.

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Step 2.3.1

Divide each term in $3 x^{2} = 4$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{4}{3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{4}{3}$

Step 2.3.2.1.2

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

$x^{2} = \frac{4}{3}$

Step 2.4

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

$x = \pm \sqrt{\frac{4}{3}}$

Step 2.5

Simplify $\pm \sqrt{\frac{4}{3}}$ .

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Step 2.5.1

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{4}}{\sqrt{3}}$

Step 2.5.2

Simplify the numerator.

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Step 2.5.2.1

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

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

Step 2.5.2.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 2.5.3

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 2.5.4

Combine and simplify the denominator.

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Step 2.5.4.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 2.5.4.2

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

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

Step 2.5.4.3

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

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

Step 2.5.4.4

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

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

Step 2.5.4.5

Add $1$ and $1$ .

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

Step 2.5.4.6

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

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Step 2.5.4.6.1

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

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

Step 2.5.4.6.2

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

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

Step 2.5.4.6.3

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

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

Step 2.5.4.6.4

Cancel the common factor of $2$ .

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Step 2.5.4.6.4.1

Cancel the common factor.

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

Step 2.5.4.6.4.2

Rewrite the expression.

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

Step 2.5.4.6.5

Evaluate the exponent.

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

Step 2.6

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

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Step 2.6.1

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

$x = \frac{2 \sqrt{3}}{3}$

Step 2.6.2

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

$x = - \frac{2 \sqrt{3}}{3}$

Step 2.6.3

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

$x = \frac{2 \sqrt{3}}{3}, - \frac{2 \sqrt{3}}{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $x^{3} - 4 x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{2 \sqrt{3}}{3}$ .

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Step 4.1.1

Substitute $\frac{2 \sqrt{3}}{3}$ for $x$ .

${(\frac{2 \sqrt{3}}{3})}^{3} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.1.1.1

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

$\frac{{(2 \sqrt{3})}^{3}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.1.2

Apply the product rule to $2 \sqrt{3}$ .

$\frac{2^{3} {\sqrt{3}}^{3}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2

Simplify the numerator.

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Step 4.1.2.1.2.1

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

$\frac{8 {\sqrt{3}}^{3}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2.2

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

$\frac{8 \sqrt{3^{3}}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2.3

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

$\frac{8 \sqrt{27}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2.4

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

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Step 4.1.2.1.2.4.1

Factor $9$ out of $27$ .

$\frac{8 \sqrt{9 (3)}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2.4.2

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

$\frac{8 \sqrt{3^{2} \cdot 3}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2.5

Pull terms out from under the radical.

$\frac{8 \cdot 3 \sqrt{3}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2.6

Multiply $8$ by $3$ .

$\frac{24 \sqrt{3}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.3

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

$\frac{24 \sqrt{3}}{27} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.4

Cancel the common factor of $24$ and $27$ .

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Step 4.1.2.1.4.1

Factor $3$ out of $24 \sqrt{3}$ .

$\frac{3 (8 \sqrt{3})}{27} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.4.2

Cancel the common factors.

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Step 4.1.2.1.4.2.1

Factor $3$ out of $27$ .

$\frac{3 (8 \sqrt{3})}{3 (9)} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.4.2.2

Cancel the common factor.

$\frac{3 (8 \sqrt{3})}{3 \cdot 9} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.4.2.3

Rewrite the expression.

$\frac{8 \sqrt{3}}{9} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.5

Multiply $- 4 \frac{2 \sqrt{3}}{3}$ .

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Step 4.1.2.1.5.1

Combine $- 4$ and $\frac{2 \sqrt{3}}{3}$ .

$\frac{8 \sqrt{3}}{9} + \frac{- 4 (2 \sqrt{3})}{3}$

Step 4.1.2.1.5.2

Multiply $2$ by $- 4$ .

$\frac{8 \sqrt{3}}{9} + \frac{- 8 \sqrt{3}}{3}$

Step 4.1.2.1.6

Move the negative in front of the fraction.

$\frac{8 \sqrt{3}}{9} - \frac{8 \sqrt{3}}{3}$

Step 4.1.2.2

To write $- \frac{8 \sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8 \sqrt{3}}{9} - \frac{8 \sqrt{3}}{3} \cdot \frac{3}{3}$

Step 4.1.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.3.1

Multiply $\frac{8 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

$\frac{8 \sqrt{3}}{9} - \frac{8 \sqrt{3} \cdot 3}{3 \cdot 3}$

Step 4.1.2.3.2

Multiply $3$ by $3$ .

$\frac{8 \sqrt{3}}{9} - \frac{8 \sqrt{3} \cdot 3}{9}$

Step 4.1.2.4

Combine the numerators over the common denominator.

$\frac{8 \sqrt{3} - 8 \sqrt{3} \cdot 3}{9}$

Step 4.1.2.5

Simplify the numerator.

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Step 4.1.2.5.1

Multiply $3$ by $- 8$ .

$\frac{8 \sqrt{3} - 24 \sqrt{3}}{9}$

Step 4.1.2.5.2

Subtract $24 \sqrt{3}$ from $8 \sqrt{3}$ .

$\frac{- 16 \sqrt{3}}{9}$

Step 4.1.2.6

Move the negative in front of the fraction.

$- \frac{16 \sqrt{3}}{9}$

Step 4.2

Evaluate at $x = - \frac{2 \sqrt{3}}{3}$ .

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Step 4.2.1

Substitute $- \frac{2 \sqrt{3}}{3}$ for $x$ .

${(- \frac{2 \sqrt{3}}{3})}^{3} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.2.1.1.1

Apply the product rule to $- \frac{2 \sqrt{3}}{3}$ .

${(- 1)}^{3} {(\frac{2 \sqrt{3}}{3})}^{3} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.1.2

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

${(- 1)}^{3} \frac{{(2 \sqrt{3})}^{3}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.1.3

Apply the product rule to $2 \sqrt{3}$ .

${(- 1)}^{3} \frac{2^{3} {\sqrt{3}}^{3}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.2

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

$- \frac{2^{3} {\sqrt{3}}^{3}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3

Simplify the numerator.

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Step 4.2.2.1.3.1

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

$- \frac{8 {\sqrt{3}}^{3}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3.2

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

$- \frac{8 \sqrt{3^{3}}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3.3

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

$- \frac{8 \sqrt{27}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3.4

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

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Step 4.2.2.1.3.4.1

Factor $9$ out of $27$ .

$- \frac{8 \sqrt{9 (3)}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3.4.2

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

$- \frac{8 \sqrt{3^{2} \cdot 3}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3.5

Pull terms out from under the radical.

$- \frac{8 \cdot 3 \sqrt{3}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3.6

Multiply $8$ by $3$ .

$- \frac{24 \sqrt{3}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.4

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

$- \frac{24 \sqrt{3}}{27} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.5

Cancel the common factor of $24$ and $27$ .

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Step 4.2.2.1.5.1

Factor $3$ out of $24 \sqrt{3}$ .

$- \frac{3 (8 \sqrt{3})}{27} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.5.2

Cancel the common factors.

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Step 4.2.2.1.5.2.1

Factor $3$ out of $27$ .

$- \frac{3 (8 \sqrt{3})}{3 (9)} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.5.2.2

Cancel the common factor.

$- \frac{3 (8 \sqrt{3})}{3 \cdot 9} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.5.2.3

Rewrite the expression.

$- \frac{8 \sqrt{3}}{9} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.6

Multiply $- 4 (- \frac{2 \sqrt{3}}{3})$ .

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Step 4.2.2.1.6.1

Multiply $- 1$ by $- 4$ .

$- \frac{8 \sqrt{3}}{9} + 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.2.1.6.2

Combine $4$ and $\frac{2 \sqrt{3}}{3}$ .

$- \frac{8 \sqrt{3}}{9} + \frac{4 (2 \sqrt{3})}{3}$

Step 4.2.2.1.6.3

Multiply $2$ by $4$ .

$- \frac{8 \sqrt{3}}{9} + \frac{8 \sqrt{3}}{3}$

Step 4.2.2.2

To write $\frac{8 \sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{8 \sqrt{3}}{9} + \frac{8 \sqrt{3}}{3} \cdot \frac{3}{3}$

Step 4.2.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.2.3.1

Multiply $\frac{8 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

$- \frac{8 \sqrt{3}}{9} + \frac{8 \sqrt{3} \cdot 3}{3 \cdot 3}$

Step 4.2.2.3.2

Multiply $3$ by $3$ .

$- \frac{8 \sqrt{3}}{9} + \frac{8 \sqrt{3} \cdot 3}{9}$

Step 4.2.2.4

Combine the numerators over the common denominator.

$\frac{- 8 \sqrt{3} + 8 \sqrt{3} \cdot 3}{9}$

Step 4.2.2.5

Simplify the numerator.

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Step 4.2.2.5.1

Multiply $3$ by $8$ .

$\frac{- 8 \sqrt{3} + 24 \sqrt{3}}{9}$

Step 4.2.2.5.2

Add $- 8 \sqrt{3}$ and $24 \sqrt{3}$ .

$\frac{16 \sqrt{3}}{9}$

Step 4.3

List all of the points.

$(\frac{2 \sqrt{3}}{3}, - \frac{16 \sqrt{3}}{9}), (- \frac{2 \sqrt{3}}{3}, \frac{16 \sqrt{3}}{9})$

Step 5