Calculus Examples

$f (x) = 2 x^{2} - \frac{6}{x^{2}}$

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

By the Sum Rule, the derivative of $2 x^{2} - \frac{6}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- \frac{6}{x^{2}}]$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

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 $n = 2$ .

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

Step 1.1.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- \frac{6}{x^{2}}]$

Step 1.1.3

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

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Step 1.1.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- \frac{6}{x^{2}}$ with respect to $x$ is $- 6 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$4 x - 6 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 1.1.3.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$4 x - 6 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 1.1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 1.1.3.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$4 x - 6 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}])$

Step 1.1.3.3.2

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

$4 x - 6 (- u^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.1.3.3.3

Replace all occurrences of $u$ with $x^{2}$ .

$4 x - 6 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.1.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$4 x - 6 (- {(x^{2})}^{- 2} (2 x))$

Step 1.1.3.5

Multiply the exponents in ${(x^{2})}^{- 2}$ .

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Step 1.1.3.5.1

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

$4 x - 6 (- x^{2 \cdot - 2} (2 x))$

Step 1.1.3.5.2

Multiply $2$ by $- 2$ .

$4 x - 6 (- x^{- 4} (2 x))$

Step 1.1.3.6

Multiply $2$ by $- 1$ .

$4 x - 6 (- 2 x^{- 4} x)$

Step 1.1.3.7

Raise $x$ to the power of $1$ .

$4 x - 6 (- 2 (x^{1} x^{- 4}))$

Step 1.1.3.8

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

$4 x - 6 (- 2 x^{1 - 4})$

Step 1.1.3.9

Subtract $4$ from $1$ .

$4 x - 6 (- 2 x^{- 3})$

Step 1.1.3.10

Multiply $- 2$ by $- 6$ .

$4 x + 12 x^{- 3}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 x + 12 \frac{1}{x^{3}}$

Step 1.1.4.2

Combine $12$ and $\frac{1}{x^{3}}$ .

$f' (x) = 4 x + \frac{12}{x^{3}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x + \frac{12}{x^{3}}$ .

$4 x + \frac{12}{x^{3}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 x + \frac{12}{x^{3}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$4 x + \frac{12}{x^{3}} = 0$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{3}, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 2.3

Multiply each term in $4 x + \frac{12}{x^{3}} = 0$ by $x^{3}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $4 x + \frac{12}{x^{3}} = 0$ by $x^{3}$ .

$4 x \cdot x^{3} + \frac{12}{x^{3}} x^{3} = 0 x^{3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.3.2.1.1.1

Move $x^{3}$ .

$4 (x^{3} x) + \frac{12}{x^{3}} x^{3} = 0 x^{3}$

Step 2.3.2.1.1.2

Multiply $x^{3}$ by $x$ .

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Step 2.3.2.1.1.2.1

Raise $x$ to the power of $1$ .

$4 (x^{3} x^{1}) + \frac{12}{x^{3}} x^{3} = 0 x^{3}$

Step 2.3.2.1.1.2.2

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

$4 x^{3 + 1} + \frac{12}{x^{3}} x^{3} = 0 x^{3}$

Step 2.3.2.1.1.3

Add $3$ and $1$ .

$4 x^{4} + \frac{12}{x^{3}} x^{3} = 0 x^{3}$

Step 2.3.2.1.2

Cancel the common factor of $x^{3}$ .

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Step 2.3.2.1.2.1

Cancel the common factor.

$4 x^{4} + \frac{12}{x^{3}} x^{3} = 0 x^{3}$

Step 2.3.2.1.2.2

Rewrite the expression.

$4 x^{4} + 12 = 0 x^{3}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0$ by $x^{3}$ .

$4 x^{4} + 12 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Subtract $12$ from both sides of the equation.

$4 x^{4} = - 12$

Step 2.4.2

Divide each term in $4 x^{4} = - 12$ by $4$ and simplify.

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Step 2.4.2.1

Divide each term in $4 x^{4} = - 12$ by $4$ .

$\frac{4 x^{4}}{4} = \frac{- 12}{4}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{4 x^{4}}{4} = \frac{- 12}{4}$

Step 2.4.2.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{- 12}{4}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $- 12$ by $4$ .

$x^{4} = - 3$

Step 2.4.3

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

$x = \pm \sqrt[4]{- 3}$

Step 2.4.4

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

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Step 2.4.4.1

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

$x = \sqrt[4]{- 3}$

Step 2.4.4.2

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

$x = - \sqrt[4]{- 3}$

Step 2.4.4.3

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

$x = \sqrt[4]{- 3}, - \sqrt[4]{- 3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{12}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

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

$x = \sqrt[3]{0}$

Step 3.2.2

Simplify $\sqrt[3]{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 3.2.2.2

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

$x = 0$

Step 4

Evaluate $2 x^{2} - \frac{6}{x^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \sqrt[4]{- 3}$ .

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Step 4.1.1

Substitute $\sqrt[4]{- 3}$ for $x$ .

$2 {(\sqrt[4]{- 3})}^{2} - \frac{6}{{(\sqrt[4]{- 3})}^{2}}$

Step 4.1.2

Simplify each term.

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Step 4.1.2.1

Rewrite ${\sqrt[4]{- 3}}^{2}$ as $\sqrt{- 3}$ .

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Step 4.1.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{- 3}$ as ${(- 3)}^{\frac{1}{4}}$ .

$2 {({(- 3)}^{\frac{1}{4}})}^{2} - \frac{6}{{(\sqrt[4]{- 3})}^{2}}$

Step 4.1.2.1.2

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

$2 {(- 3)}^{\frac{1}{4} \cdot 2} - \frac{6}{{(\sqrt[4]{- 3})}^{2}}$

Step 4.1.2.1.3

Combine $\frac{1}{4}$ and $2$ .

$2 {(- 3)}^{\frac{2}{4}} - \frac{6}{{(\sqrt[4]{- 3})}^{2}}$

Step 4.1.2.1.4

Cancel the common factor of $2$ and $4$ .

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Step 4.1.2.1.4.1

Factor $2$ out of $2$ .

$2 {(- 3)}^{\frac{2 (1)}{4}} - \frac{6}{{(\sqrt[4]{- 3})}^{2}}$

Step 4.1.2.1.4.2

Cancel the common factors.

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Step 4.1.2.1.4.2.1

Factor $2$ out of $4$ .

$2 {(- 3)}^{\frac{2 \cdot 1}{2 \cdot 2}} - \frac{6}{{(\sqrt[4]{- 3})}^{2}}$

Step 4.1.2.1.4.2.2

Cancel the common factor.

$2 {(- 3)}^{\frac{2 \cdot 1}{2 \cdot 2}} - \frac{6}{{(\sqrt[4]{- 3})}^{2}}$

Step 4.1.2.1.4.2.3

Rewrite the expression.

$2 {(- 3)}^{\frac{1}{2}} - \frac{6}{{(\sqrt[4]{- 3})}^{2}}$

Step 4.1.2.1.5

Rewrite ${(- 3)}^{\frac{1}{2}}$ as $\sqrt{- 3}$ .

$2 \sqrt{- 3} - \frac{6}{{(\sqrt[4]{- 3})}^{2}}$

Step 4.1.2.2

Rewrite $- 3$ as $- 1 (3)$ .

$2 \sqrt{- 1 (3)} - \frac{6}{{(\sqrt[4]{- 3})}^{2}}$

Step 4.1.2.3

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$2 (\sqrt{- 1} \cdot \sqrt{3}) - \frac{6}{{(\sqrt[4]{- 3})}^{2}}$

Step 4.1.2.4

Rewrite $\sqrt{- 1}$ as $i$ .

$2 (i \cdot \sqrt{3}) - \frac{6}{{(\sqrt[4]{- 3})}^{2}}$

Step 4.1.2.5

Simplify the denominator.

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Step 4.1.2.5.1

Rewrite ${\sqrt[4]{- 3}}^{2}$ as $\sqrt{- 3}$ .

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Step 4.1.2.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{- 3}$ as ${(- 3)}^{\frac{1}{4}}$ .

$2 i \sqrt{3} - \frac{6}{{({(- 3)}^{\frac{1}{4}})}^{2}}$

Step 4.1.2.5.1.2

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

$2 i \sqrt{3} - \frac{6}{{(- 3)}^{\frac{1}{4} \cdot 2}}$

Step 4.1.2.5.1.3

Combine $\frac{1}{4}$ and $2$ .

$2 i \sqrt{3} - \frac{6}{{(- 3)}^{\frac{2}{4}}}$

Step 4.1.2.5.1.4

Cancel the common factor of $2$ and $4$ .

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Step 4.1.2.5.1.4.1

Factor $2$ out of $2$ .

$2 i \sqrt{3} - \frac{6}{{(- 3)}^{\frac{2 (1)}{4}}}$

Step 4.1.2.5.1.4.2

Cancel the common factors.

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Step 4.1.2.5.1.4.2.1

Factor $2$ out of $4$ .

$2 i \sqrt{3} - \frac{6}{{(- 3)}^{\frac{2 \cdot 1}{2 \cdot 2}}}$

Step 4.1.2.5.1.4.2.2

Cancel the common factor.

$2 i \sqrt{3} - \frac{6}{{(- 3)}^{\frac{2 \cdot 1}{2 \cdot 2}}}$

Step 4.1.2.5.1.4.2.3

Rewrite the expression.

$2 i \sqrt{3} - \frac{6}{{(- 3)}^{\frac{1}{2}}}$

Step 4.1.2.5.1.5

Rewrite ${(- 3)}^{\frac{1}{2}}$ as $\sqrt{- 3}$ .

$2 i \sqrt{3} - \frac{6}{\sqrt{- 3}}$

Step 4.1.2.5.2

Rewrite $- 3$ as $- 1 (3)$ .

$2 i \sqrt{3} - \frac{6}{\sqrt{- 1 (3)}}$

Step 4.1.2.5.3

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$2 i \sqrt{3} - \frac{6}{\sqrt{- 1} \cdot \sqrt{3}}$

Step 4.1.2.5.4

Rewrite $\sqrt{- 1}$ as $i$ .

$2 i \sqrt{3} - \frac{6}{i \sqrt{3}}$

Step 4.1.2.6

Multiply the numerator and denominator of $\frac{6}{1.7320508 i}$ by the conjugate of $1.7320508 i$ to make the denominator real.

$2 i \sqrt{3} - (\frac{6}{1.7320508 i} \cdot \frac{i}{i})$

Step 4.1.2.7

Multiply.

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Step 4.1.2.7.1

Combine.

$2 i \sqrt{3} - \frac{6 i}{1.7320508 i i}$

Step 4.1.2.7.2

Simplify the denominator.

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Step 4.1.2.7.2.1

Add parentheses.

$2 i \sqrt{3} - \frac{6 i}{1.7320508 (i i)}$

Step 4.1.2.7.2.2

Raise $i$ to the power of $1$ .

$2 i \sqrt{3} - \frac{6 i}{1.7320508 (i^{1} i)}$

Step 4.1.2.7.2.3

Raise $i$ to the power of $1$ .

$2 i \sqrt{3} - \frac{6 i}{1.7320508 (i^{1} i^{1})}$

Step 4.1.2.7.2.4

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

$2 i \sqrt{3} - \frac{6 i}{1.7320508 i^{1 + 1}}$

Step 4.1.2.7.2.5

Add $1$ and $1$ .

$2 i \sqrt{3} - \frac{6 i}{1.7320508 i^{2}}$

Step 4.1.2.7.2.6

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

$2 i \sqrt{3} - \frac{6 i}{1.7320508 \cdot - 1}$

Step 4.1.2.8

Multiply $1.7320508$ by $- 1$ .

$2 i \sqrt{3} - \frac{6 i}{- 1.7320508}$

Step 4.1.2.9

Move the negative in front of the fraction.

$2 i \sqrt{3} - - \frac{6 i}{1.7320508}$

Step 4.1.2.10

Factor $6$ out of $6 i$ .

$2 i \sqrt{3} - - \frac{6 (i)}{1.7320508}$

Step 4.1.2.11

Factor $1.7320508$ out of $1.7320508$ .

$2 i \sqrt{3} - - \frac{6 (i)}{1.7320508 (1)}$

Step 4.1.2.12

Separate fractions.

$2 i \sqrt{3} - - (\frac{6}{1.7320508} \cdot \frac{i}{1})$

Step 4.1.2.13

Divide $6$ by $1.7320508$ .

$2 i \sqrt{3} - - (3.46410161 \frac{i}{1})$

Step 4.1.2.14

Divide $i$ by $1$ .

$2 i \sqrt{3} - - (3.46410161 i)$

Step 4.1.2.15

Multiply $3.46410161$ by $- 1$ .

$2 i \sqrt{3} - (- 3.46410161 i)$

Step 4.1.2.16

Multiply $- 3.46410161$ by $- 1$ .

$2 i \sqrt{3} + 3.46410161 i$

Step 4.2

Evaluate at $x = - \sqrt[4]{- 3}$ .

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Step 4.2.1

Substitute $- \sqrt[4]{- 3}$ for $x$ .

$2 {(- \sqrt[4]{- 3})}^{2} - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Apply the product rule to $- \sqrt[4]{- 3}$ .

$2 ({(- 1)}^{2} {\sqrt[4]{- 3}}^{2}) - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.2

Raise $- 1$ to the power of $2$ .

$2 (1 {\sqrt[4]{- 3}}^{2}) - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.3

Multiply ${\sqrt[4]{- 3}}^{2}$ by $1$ .

$2 {\sqrt[4]{- 3}}^{2} - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.4

Rewrite ${\sqrt[4]{- 3}}^{2}$ as $\sqrt{- 3}$ .

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Step 4.2.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{- 3}$ as ${(- 3)}^{\frac{1}{4}}$ .

$2 {({(- 3)}^{\frac{1}{4}})}^{2} - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.4.2

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

$2 {(- 3)}^{\frac{1}{4} \cdot 2} - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.4.3

Combine $\frac{1}{4}$ and $2$ .

$2 {(- 3)}^{\frac{2}{4}} - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.4.4

Cancel the common factor of $2$ and $4$ .

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Step 4.2.2.4.4.1

Factor $2$ out of $2$ .

$2 {(- 3)}^{\frac{2 (1)}{4}} - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.4.4.2

Cancel the common factors.

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Step 4.2.2.4.4.2.1

Factor $2$ out of $4$ .

$2 {(- 3)}^{\frac{2 \cdot 1}{2 \cdot 2}} - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.4.4.2.2

Cancel the common factor.

$2 {(- 3)}^{\frac{2 \cdot 1}{2 \cdot 2}} - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.4.4.2.3

Rewrite the expression.

$2 {(- 3)}^{\frac{1}{2}} - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.4.5

Rewrite ${(- 3)}^{\frac{1}{2}}$ as $\sqrt{- 3}$ .

$2 \sqrt{- 3} - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.5

Rewrite $- 3$ as $- 1 (3)$ .

$2 \sqrt{- 1 (3)} - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.6

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$2 (\sqrt{- 1} \cdot \sqrt{3}) - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.7

Rewrite $\sqrt{- 1}$ as $i$ .

$2 (i \cdot \sqrt{3}) - \frac{6}{{(- \sqrt[4]{- 3})}^{2}}$

Step 4.2.2.8

Simplify the denominator.

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Step 4.2.2.8.1

Apply the product rule to $- \sqrt[4]{- 3}$ .

$2 i \sqrt{3} - \frac{6}{{(- 1)}^{2} {\sqrt[4]{- 3}}^{2}}$

Step 4.2.2.8.2

Raise $- 1$ to the power of $2$ .

$2 i \sqrt{3} - \frac{6}{1 {\sqrt[4]{- 3}}^{2}}$

Step 4.2.2.8.3

Rewrite ${\sqrt[4]{- 3}}^{2}$ as $\sqrt{- 3}$ .

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Step 4.2.2.8.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{- 3}$ as ${(- 3)}^{\frac{1}{4}}$ .

$2 i \sqrt{3} - \frac{6}{1 {({(- 3)}^{\frac{1}{4}})}^{2}}$

Step 4.2.2.8.3.2

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

$2 i \sqrt{3} - \frac{6}{1 {(- 3)}^{\frac{1}{4} \cdot 2}}$

Step 4.2.2.8.3.3

Combine $\frac{1}{4}$ and $2$ .

$2 i \sqrt{3} - \frac{6}{1 {(- 3)}^{\frac{2}{4}}}$

Step 4.2.2.8.3.4

Cancel the common factor of $2$ and $4$ .

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Step 4.2.2.8.3.4.1

Factor $2$ out of $2$ .

$2 i \sqrt{3} - \frac{6}{1 {(- 3)}^{\frac{2 (1)}{4}}}$

Step 4.2.2.8.3.4.2

Cancel the common factors.

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Step 4.2.2.8.3.4.2.1

Factor $2$ out of $4$ .

$2 i \sqrt{3} - \frac{6}{1 {(- 3)}^{\frac{2 \cdot 1}{2 \cdot 2}}}$

Step 4.2.2.8.3.4.2.2

Cancel the common factor.

$2 i \sqrt{3} - \frac{6}{1 {(- 3)}^{\frac{2 \cdot 1}{2 \cdot 2}}}$

Step 4.2.2.8.3.4.2.3

Rewrite the expression.

$2 i \sqrt{3} - \frac{6}{1 {(- 3)}^{\frac{1}{2}}}$

Step 4.2.2.8.3.5

Rewrite ${(- 3)}^{\frac{1}{2}}$ as $\sqrt{- 3}$ .

$2 i \sqrt{3} - \frac{6}{1 \sqrt{- 3}}$

Step 4.2.2.8.4

Rewrite $- 3$ as $- 1 (3)$ .

$2 i \sqrt{3} - \frac{6}{1 \sqrt{- 1 (3)}}$

Step 4.2.2.8.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$2 i \sqrt{3} - \frac{6}{1 (\sqrt{- 1} \cdot \sqrt{3})}$

Step 4.2.2.8.6

Rewrite $\sqrt{- 1}$ as $i$ .

$2 i \sqrt{3} - \frac{6}{1 (i \cdot \sqrt{3})}$

Step 4.2.2.8.7

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

$2 i \sqrt{3} - \frac{6}{i \sqrt{3}}$

Step 4.2.2.9

Multiply the numerator and denominator of $\frac{6}{1.7320508 i}$ by the conjugate of $1.7320508 i$ to make the denominator real.

$2 i \sqrt{3} - (\frac{6}{1.7320508 i} \cdot \frac{i}{i})$

Step 4.2.2.10

Multiply.

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Step 4.2.2.10.1

Combine.

$2 i \sqrt{3} - \frac{6 i}{1.7320508 i i}$

Step 4.2.2.10.2

Simplify the denominator.

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Step 4.2.2.10.2.1

Add parentheses.

$2 i \sqrt{3} - \frac{6 i}{1.7320508 (i i)}$

Step 4.2.2.10.2.2

Raise $i$ to the power of $1$ .

$2 i \sqrt{3} - \frac{6 i}{1.7320508 (i^{1} i)}$

Step 4.2.2.10.2.3

Raise $i$ to the power of $1$ .

$2 i \sqrt{3} - \frac{6 i}{1.7320508 (i^{1} i^{1})}$

Step 4.2.2.10.2.4

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

$2 i \sqrt{3} - \frac{6 i}{1.7320508 i^{1 + 1}}$

Step 4.2.2.10.2.5

Add $1$ and $1$ .

$2 i \sqrt{3} - \frac{6 i}{1.7320508 i^{2}}$

Step 4.2.2.10.2.6

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

$2 i \sqrt{3} - \frac{6 i}{1.7320508 \cdot - 1}$

Step 4.2.2.11

Multiply $1.7320508$ by $- 1$ .

$2 i \sqrt{3} - \frac{6 i}{- 1.7320508}$

Step 4.2.2.12

Move the negative in front of the fraction.

$2 i \sqrt{3} - - \frac{6 i}{1.7320508}$

Step 4.2.2.13

Factor $6$ out of $6 i$ .

$2 i \sqrt{3} - - \frac{6 (i)}{1.7320508}$

Step 4.2.2.14

Factor $1.7320508$ out of $1.7320508$ .

$2 i \sqrt{3} - - \frac{6 (i)}{1.7320508 (1)}$

Step 4.2.2.15

Separate fractions.

$2 i \sqrt{3} - - (\frac{6}{1.7320508} \cdot \frac{i}{1})$

Step 4.2.2.16

Divide $6$ by $1.7320508$ .

$2 i \sqrt{3} - - (3.46410161 \frac{i}{1})$

Step 4.2.2.17

Divide $i$ by $1$ .

$2 i \sqrt{3} - - (3.46410161 i)$

Step 4.2.2.18

Multiply $3.46410161$ by $- 1$ .

$2 i \sqrt{3} - (- 3.46410161 i)$

Step 4.2.2.19

Multiply $- 3.46410161$ by $- 1$ .

$2 i \sqrt{3} + 3.46410161 i$

Step 4.3

Evaluate at $x = 0$ .

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Step 4.3.1

Substitute $0$ for $x$ .

$2 {(0)}^{2} - \frac{6}{{(0)}^{2}}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Raising $0$ to any positive power yields $0$ .

$2 {(0)}^{2} - \frac{6}{0}$

Step 4.3.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 5

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found