Calculus Examples

$2 x - \frac{2}{x^{2}}$

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 $2 x - \frac{2}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{2}{x^{2}}]$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

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 $n = 1$ .

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

Step 1.1.2.3

Multiply $2$ by $1$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

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 $x^{2}$ .

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

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 $n = - 1$ .

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

Step 1.1.3.3.3

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

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

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 $n = 2$ .

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.1.3.5.2

Multiply $2$ by $- 2$ .

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

Step 1.1.3.6

Multiply $2$ by $- 1$ .

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

Step 1.1.3.7

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

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

Step 1.1.3.8

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

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

Step 1.1.3.9

Subtract $4$ from $1$ .

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

Step 1.1.3.10

Multiply $- 2$ by $- 2$ .

$2 + 4 x^{- 3}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

$2 + 4 \frac{1}{x^{3}}$

Step 1.1.4.2

Combine $4$ and $\frac{1}{x^{3}}$ .

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

Step 1.1.4.3

Reorder terms.

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{4}{x^{3}} + 2 = 0$

Step 2.2

Subtract $2$ from both sides of the equation.

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

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

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

$x^{3}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 2.4

Multiply each term in $\frac{4}{x^{3}} = - 2$ by $x^{3}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{4}{x^{3}} = - 2$ by $x^{3}$ .

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

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

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Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Rewrite the expression.

$4 = - 2 x^{3}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $- 2 x^{3} = 4$ .

$- 2 x^{3} = 4$

Step 2.5.2

Subtract $4$ from both sides of the equation.

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

Step 2.5.3

Factor $- 2$ out of $- 2 x^{3} - 4$ .

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Step 2.5.3.1

Factor $- 2$ out of $- 2 x^{3}$ .

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

Step 2.5.3.2

Factor $- 2$ out of $- 4$ .

$- 2 x^{3} - 2 \cdot 2 = 0$

Step 2.5.3.3

Factor $- 2$ out of $- 2 (x^{3}) - 2 (2)$ .

$- 2 (x^{3} + 2) = 0$

Step 2.5.4

Divide each term in $- 2 (x^{3} + 2) = 0$ by $- 2$ and simplify.

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Step 2.5.4.1

Divide each term in $- 2 (x^{3} + 2) = 0$ by $- 2$ .

$\frac{- 2 (x^{3} + 2)}{- 2} = \frac{0}{- 2}$

Step 2.5.4.2

Simplify the left side.

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Step 2.5.4.2.1

Cancel the common factor of $- 2$ .

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Step 2.5.4.2.1.1

Cancel the common factor.

$\frac{- 2 (x^{3} + 2)}{- 2} = \frac{0}{- 2}$

Step 2.5.4.2.1.2

Divide $x^{3} + 2$ by $1$ .

$x^{3} + 2 = \frac{0}{- 2}$

Step 2.5.4.3

Simplify the right side.

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Step 2.5.4.3.1

Divide $0$ by $- 2$ .

$x^{3} + 2 = 0$

Step 2.5.5

Subtract $2$ from both sides of the equation.

$x^{3} = - 2$

Step 2.5.6

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

$x = \sqrt[3]{- 2}$

Step 2.5.7

Simplify $\sqrt[3]{- 2}$ .

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Step 2.5.7.1

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

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Step 2.5.7.1.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \sqrt[3]{- 1 (2)}$

Step 2.5.7.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \cdot 2}$

Step 2.5.7.2

Pull terms out from under the radical.

$x = - 1 \sqrt[3]{2}$

Step 2.5.7.3

Rewrite $- 1 \sqrt[3]{2}$ as $- \sqrt[3]{2}$ .

$x = - \sqrt[3]{2}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{4}{x^{3}}$ 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 $2 x - \frac{2}{x^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

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

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Step 4.1.1

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

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

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

Multiply $- 1$ by $2$ .

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

Step 4.1.2.1.2

Simplify the denominator.

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Step 4.1.2.1.2.1

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

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

Step 4.1.2.1.2.2

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

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

Step 4.1.2.1.2.3

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

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

Step 4.1.2.1.2.4

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

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

Step 4.1.2.1.2.5

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

$- 2 \sqrt[3]{2} - \frac{2}{\sqrt[3]{4}}$

Step 4.1.2.1.3

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

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

Step 4.1.2.1.4

Combine and simplify the denominator.

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Step 4.1.2.1.4.1

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

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

Step 4.1.2.1.4.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

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

Step 4.1.2.1.4.3

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

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

Step 4.1.2.1.4.4

Add $1$ and $2$ .

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

Step 4.1.2.1.4.5

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

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Step 4.1.2.1.4.5.1

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

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

Step 4.1.2.1.4.5.2

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

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

Step 4.1.2.1.4.5.3

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

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

Step 4.1.2.1.4.5.4

Cancel the common factor of $3$ .

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Step 4.1.2.1.4.5.4.1

Cancel the common factor.

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

Step 4.1.2.1.4.5.4.2

Rewrite the expression.

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

Step 4.1.2.1.4.5.5

Evaluate the exponent.

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

Step 4.1.2.1.5

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

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Step 4.1.2.1.5.1

Factor $2$ out of $2 {\sqrt[3]{4}}^{2}$ .

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

Step 4.1.2.1.5.2

Cancel the common factors.

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Step 4.1.2.1.5.2.1

Factor $2$ out of $4$ .

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

Step 4.1.2.1.5.2.2

Cancel the common factor.

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

Step 4.1.2.1.5.2.3

Rewrite the expression.

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

Step 4.1.2.1.6

Simplify the numerator.

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Step 4.1.2.1.6.1

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

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

Step 4.1.2.1.6.2

Raise $4$ to the power of $2$ .

$- 2 \sqrt[3]{2} - \frac{\sqrt[3]{16}}{2}$

Step 4.1.2.1.6.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 4.1.2.1.6.3.1

Factor $8$ out of $16$ .

$- 2 \sqrt[3]{2} - \frac{\sqrt[3]{8 (2)}}{2}$

Step 4.1.2.1.6.3.2

Rewrite $8$ as $2^{3}$ .

$- 2 \sqrt[3]{2} - \frac{\sqrt[3]{2^{3} \cdot 2}}{2}$

Step 4.1.2.1.6.4

Pull terms out from under the radical.

$- 2 \sqrt[3]{2} - \frac{2 \sqrt[3]{2}}{2}$

Step 4.1.2.1.7

Cancel the common factor of $2$ .

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Step 4.1.2.1.7.1

Cancel the common factor.

$- 2 \sqrt[3]{2} - \frac{2 \sqrt[3]{2}}{2}$

Step 4.1.2.1.7.2

Divide $\sqrt[3]{2}$ by $1$ .

$- 2 \sqrt[3]{2} - \sqrt[3]{2}$

Step 4.1.2.2

Subtract $\sqrt[3]{2}$ from $- 2 \sqrt[3]{2}$ .

$- 3 \sqrt[3]{2}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

Undefined

Step 4.3

List all of the points.

$(- \sqrt[3]{2}, - 3 \sqrt[3]{2})$

Step 5