Calculus Examples

$f (x) = x^{2} + \frac{120}{x}$

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 $x^{2} + \frac{120}{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{120}{x}]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{120}{x}]$

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 $n = 2$ .

$2 x + \frac{d}{d x} [\frac{120}{x}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{120}{x}]$ .

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Step 1.1.2.1

Since $120$ is constant with respect to $x$ , the derivative of $\frac{120}{x}$ with respect to $x$ is $120 \frac{d}{d x} [\frac{1}{x}]$ .

$2 x + 120 \frac{d}{d x} [\frac{1}{x}]$

Step 1.1.2.2

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

$2 x + 120 \frac{d}{d x} [x^{- 1}]$

Step 1.1.2.3

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

$2 x + 120 (- x^{- 2})$

Step 1.1.2.4

Multiply $- 1$ by $120$ .

$2 x - 120 x^{- 2}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

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

$2 x - 120 \frac{1}{x^{2}}$

Step 1.1.3.2

Combine terms.

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Step 1.1.3.2.1

Combine $- 120$ and $\frac{1}{x^{2}}$ .

$2 x + \frac{- 120}{x^{2}}$

Step 1.1.3.2.2

Move the negative in front of the fraction.

$f' (x) = 2 x - \frac{120}{x^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 x - \frac{120}{x^{2}}$ .

$2 x - \frac{120}{x^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $2 x - \frac{120}{x^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$2 x - \frac{120}{x^{2}} = 0$

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.

$1, x^{2}, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 2.3

Multiply each term in $2 x - \frac{120}{x^{2}} = 0$ by $x^{2}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $2 x - \frac{120}{x^{2}} = 0$ by $x^{2}$ .

$2 x \cdot x^{2} - \frac{120}{x^{2}} x^{2} = 0 x^{2}$

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 $x^{2}$ by adding the exponents.

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Step 2.3.2.1.1.1

Move $x^{2}$ .

$2 (x^{2} x) - \frac{120}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.1.2

Multiply $x^{2}$ by $x$ .

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Step 2.3.2.1.1.2.1

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

$2 (x^{2} x^{1}) - \frac{120}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.1.2.2

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

$2 x^{2 + 1} - \frac{120}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.1.3

Add $2$ and $1$ .

$2 x^{3} - \frac{120}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.2

Cancel the common factor of $x^{2}$ .

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Step 2.3.2.1.2.1

Move the leading negative in $- \frac{120}{x^{2}}$ into the numerator.

$2 x^{3} + \frac{- 120}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.2.2

Cancel the common factor.

$2 x^{3} + \frac{- 120}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.2.3

Rewrite the expression.

$2 x^{3} - 120 = 0 x^{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0$ by $x^{2}$ .

$2 x^{3} - 120 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Add $120$ to both sides of the equation.

$2 x^{3} = 120$

Step 2.4.2

Subtract $120$ from both sides of the equation.

$2 x^{3} - 120 = 0$

Step 2.4.3

Factor $2$ out of $2 x^{3} - 120$ .

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Step 2.4.3.1

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

$2 (x^{3}) - 120 = 0$

Step 2.4.3.2

Factor $2$ out of $- 120$ .

$2 x^{3} + 2 \cdot - 60 = 0$

Step 2.4.3.3

Factor $2$ out of $2 x^{3} + 2 \cdot - 60$ .

$2 (x^{3} - 60) = 0$

Step 2.4.4

Divide each term in $2 (x^{3} - 60) = 0$ by $2$ and simplify.

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Step 2.4.4.1

Divide each term in $2 (x^{3} - 60) = 0$ by $2$ .

$\frac{2 (x^{3} - 60)}{2} = \frac{0}{2}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $2$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{2 (x^{3} - 60)}{2} = \frac{0}{2}$

Step 2.4.4.2.1.2

Divide $x^{3} - 60$ by $1$ .

$x^{3} - 60 = \frac{0}{2}$

Step 2.4.4.3

Simplify the right side.

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Step 2.4.4.3.1

Divide $0$ by $2$ .

$x^{3} - 60 = 0$

Step 2.4.5

Add $60$ to both sides of the equation.

$x^{3} = 60$

Step 2.4.6

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

$x = \sqrt[3]{60}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{120}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

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.

$x = \pm \sqrt{0}$

Step 3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.2.2.2

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

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Evaluate $x^{2} + \frac{120}{x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \sqrt[3]{60}$ .

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Step 4.1.1

Substitute $\sqrt[3]{60}$ for $x$ .

${(\sqrt[3]{60})}^{2} + \frac{120}{\sqrt[3]{60}}$

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

Rewrite ${\sqrt[3]{60}}^{2}$ as $\sqrt[3]{60^{2}}$ .

$\sqrt[3]{60^{2}} + \frac{120}{\sqrt[3]{60}}$

Step 4.1.2.1.2

Raise $60$ to the power of $2$ .

$\sqrt[3]{3600} + \frac{120}{\sqrt[3]{60}}$

Step 4.1.2.1.3

Rewrite $3600$ as $2^{3} \cdot 450$ .

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Step 4.1.2.1.3.1

Factor $8$ out of $3600$ .

$\sqrt[3]{8 (450)} + \frac{120}{\sqrt[3]{60}}$

Step 4.1.2.1.3.2

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

$\sqrt[3]{2^{3} \cdot 450} + \frac{120}{\sqrt[3]{60}}$

Step 4.1.2.1.4

Pull terms out from under the radical.

$2 \sqrt[3]{450} + \frac{120}{\sqrt[3]{60}}$

Step 4.1.2.1.5

Multiply $\frac{120}{\sqrt[3]{60}}$ by $\frac{{\sqrt[3]{60}}^{2}}{{\sqrt[3]{60}}^{2}}$ .

$2 \sqrt[3]{450} + \frac{120}{\sqrt[3]{60}} \cdot \frac{{\sqrt[3]{60}}^{2}}{{\sqrt[3]{60}}^{2}}$

Step 4.1.2.1.6

Combine and simplify the denominator.

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Step 4.1.2.1.6.1

Multiply $\frac{120}{\sqrt[3]{60}}$ by $\frac{{\sqrt[3]{60}}^{2}}{{\sqrt[3]{60}}^{2}}$ .

$2 \sqrt[3]{450} + \frac{120 {\sqrt[3]{60}}^{2}}{\sqrt[3]{60} {\sqrt[3]{60}}^{2}}$

Step 4.1.2.1.6.2

Raise $\sqrt[3]{60}$ to the power of $1$ .

$2 \sqrt[3]{450} + \frac{120 {\sqrt[3]{60}}^{2}}{{\sqrt[3]{60}}^{1} {\sqrt[3]{60}}^{2}}$

Step 4.1.2.1.6.3

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

$2 \sqrt[3]{450} + \frac{120 {\sqrt[3]{60}}^{2}}{{\sqrt[3]{60}}^{1 + 2}}$

Step 4.1.2.1.6.4

Add $1$ and $2$ .

$2 \sqrt[3]{450} + \frac{120 {\sqrt[3]{60}}^{2}}{{\sqrt[3]{60}}^{3}}$

Step 4.1.2.1.6.5

Rewrite ${\sqrt[3]{60}}^{3}$ as $60$ .

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Step 4.1.2.1.6.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{60}$ as $60^{\frac{1}{3}}$ .

$2 \sqrt[3]{450} + \frac{120 {\sqrt[3]{60}}^{2}}{{(60^{\frac{1}{3}})}^{3}}$

Step 4.1.2.1.6.5.2

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

$2 \sqrt[3]{450} + \frac{120 {\sqrt[3]{60}}^{2}}{60^{\frac{1}{3} \cdot 3}}$

Step 4.1.2.1.6.5.3

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

$2 \sqrt[3]{450} + \frac{120 {\sqrt[3]{60}}^{2}}{60^{\frac{3}{3}}}$

Step 4.1.2.1.6.5.4

Cancel the common factor of $3$ .

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Step 4.1.2.1.6.5.4.1

Cancel the common factor.

$2 \sqrt[3]{450} + \frac{120 {\sqrt[3]{60}}^{2}}{60^{\frac{3}{3}}}$

Step 4.1.2.1.6.5.4.2

Rewrite the expression.

$2 \sqrt[3]{450} + \frac{120 {\sqrt[3]{60}}^{2}}{60^{1}}$

Step 4.1.2.1.6.5.5

Evaluate the exponent.

$2 \sqrt[3]{450} + \frac{120 {\sqrt[3]{60}}^{2}}{60}$

Step 4.1.2.1.7

Cancel the common factor of $120$ and $60$ .

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Step 4.1.2.1.7.1

Factor $60$ out of $120 {\sqrt[3]{60}}^{2}$ .

$2 \sqrt[3]{450} + \frac{60 (2 {\sqrt[3]{60}}^{2})}{60}$

Step 4.1.2.1.7.2

Cancel the common factors.

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Step 4.1.2.1.7.2.1

Factor $60$ out of $60$ .

$2 \sqrt[3]{450} + \frac{60 (2 {\sqrt[3]{60}}^{2})}{60 (1)}$

Step 4.1.2.1.7.2.2

Cancel the common factor.

$2 \sqrt[3]{450} + \frac{60 (2 {\sqrt[3]{60}}^{2})}{60 \cdot 1}$

Step 4.1.2.1.7.2.3

Rewrite the expression.

$2 \sqrt[3]{450} + \frac{2 {\sqrt[3]{60}}^{2}}{1}$

Step 4.1.2.1.7.2.4

Divide $2 {\sqrt[3]{60}}^{2}$ by $1$ .

$2 \sqrt[3]{450} + 2 {\sqrt[3]{60}}^{2}$

Step 4.1.2.1.8

Rewrite ${\sqrt[3]{60}}^{2}$ as $\sqrt[3]{60^{2}}$ .

$2 \sqrt[3]{450} + 2 \sqrt[3]{60^{2}}$

Step 4.1.2.1.9

Raise $60$ to the power of $2$ .

$2 \sqrt[3]{450} + 2 \sqrt[3]{3600}$

Step 4.1.2.1.10

Rewrite $3600$ as $2^{3} \cdot 450$ .

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Step 4.1.2.1.10.1

Factor $8$ out of $3600$ .

$2 \sqrt[3]{450} + 2 \sqrt[3]{8 (450)}$

Step 4.1.2.1.10.2

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

$2 \sqrt[3]{450} + 2 \sqrt[3]{2^{3} \cdot 450}$

Step 4.1.2.1.11

Pull terms out from under the radical.

$2 \sqrt[3]{450} + 2 (2 \sqrt[3]{450})$

Step 4.1.2.1.12

Multiply $2$ by $2$ .

$2 \sqrt[3]{450} + 4 \sqrt[3]{450}$

Step 4.1.2.2

Add $2 \sqrt[3]{450}$ and $4 \sqrt[3]{450}$ .

$6 \sqrt[3]{450}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

${(0)}^{2} + \frac{120}{0}$

Step 4.2.2

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

Undefined

Step 4.3

List all of the points.

$(\sqrt[3]{60}, 6 \sqrt[3]{450})$

Step 5