Calculus Examples

$f (x) = x \sqrt{30 - x^{2}}$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{30 - x^{2}}$ as ${(30 - x^{2})}^{\frac{1}{2}}$ .

$f' (x) = \frac{d}{d x} (x {(30 - x^{2})}^{\frac{1}{2}})$

Step 1.1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(30 - x^{2})}^{\frac{1}{2}}$ .

$f' (x) = x \frac{d}{d x} ({(30 - x^{2})}^{\frac{1}{2}}) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 30 - x^{2}$ .

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Step 1.1.3.1

To apply the Chain Rule, set $u$ as $30 - x^{2}$ .

$f' (x) = x (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (30 - x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.3.2

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

$f' (x) = x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (30 - x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.3.3

Replace all occurrences of $u$ with $30 - x^{2}$ .

$f' (x) = x (\frac{1}{2} \cdot {(30 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} (30 - x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (x) = x (\frac{1}{2} \cdot {(30 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (30 - x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.5

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

$f' (x) = x (\frac{1}{2} \cdot {(30 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (30 - x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.6

Combine the numerators over the common denominator.

$f' (x) = x (\frac{1}{2} \cdot {(30 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (30 - x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Multiply $- 1$ by $2$ .

$f' (x) = x (\frac{1}{2} \cdot {(30 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} (30 - x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.7.2

Subtract $2$ from $1$ .

$f' (x) = x (\frac{1}{2} \cdot {(30 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} (30 - x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.8

Combine fractions.

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Step 1.1.8.1

Move the negative in front of the fraction.

$f' (x) = x (\frac{1}{2} \cdot {(30 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} (30 - x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.8.2

Combine $\frac{1}{2}$ and ${(30 - x^{2})}^{- \frac{1}{2}}$ .

$f' (x) = x (\frac{{(30 - x^{2})}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (30 - x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.8.3

Move ${(30 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = x (\frac{1}{2 {(30 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (30 - x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.8.4

Combine $\frac{1}{2 {(30 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$f' (x) = \frac{x}{2 {(30 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (30 - x^{2}) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.9

By the Sum Rule, the derivative of $30 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [30] + \frac{d}{d x} [- x^{2}]$ .

$f' (x) = \frac{x}{2 {(30 - x^{2})}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (30) + \frac{d}{d x} (- x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.10

Since $30$ is constant with respect to $x$ , the derivative of $30$ with respect to $x$ is $0$ .

$f' (x) = \frac{x}{2 {(30 - x^{2})}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (- x^{2})) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.11

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$f' (x) = \frac{x}{2 {(30 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x^{2}) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.12

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

$f' (x) = \frac{x}{2 {(30 - x^{2})}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x^{2}) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.13

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

$f' (x) = \frac{x}{2 {(30 - x^{2})}^{\frac{1}{2}}} \cdot (- (2 x)) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.14

Combine fractions.

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Step 1.1.14.1

Multiply $2$ by $- 1$ .

$f' (x) = \frac{x}{2 {(30 - x^{2})}^{\frac{1}{2}}} \cdot (- 2 x) + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.14.2

Combine $- 2$ and $\frac{x}{2 {(30 - x^{2})}^{\frac{1}{2}}}$ .

$f' (x) = \frac{- 2 x}{2 {(30 - x^{2})}^{\frac{1}{2}}} \cdot x + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.14.3

Combine $\frac{- 2 x}{2 {(30 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$f' (x) = \frac{- 2 x \cdot x}{2 {(30 - x^{2})}^{\frac{1}{2}}} + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.15

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

$f' (x) = \frac{- 2 (x \cdot x)}{2 {(30 - x^{2})}^{\frac{1}{2}}} + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.16

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

$f' (x) = \frac{- 2 (x \cdot x)}{2 {(30 - x^{2})}^{\frac{1}{2}}} + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.17

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

$f' (x) = \frac{- 2 x^{1 + 1}}{2 {(30 - x^{2})}^{\frac{1}{2}}} + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.18

Add $1$ and $1$ .

$f' (x) = \frac{- 2 x^{2}}{2 {(30 - x^{2})}^{\frac{1}{2}}} + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.19

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

$f' (x) = \frac{2 (- x^{2})}{2 {(30 - x^{2})}^{\frac{1}{2}}} + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.20

Cancel the common factors.

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Step 1.1.20.1

Factor $2$ out of $2 {(30 - x^{2})}^{\frac{1}{2}}$ .

$f' (x) = \frac{2 (- x^{2})}{2 ({(30 - x^{2})}^{\frac{1}{2}})} + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.20.2

Cancel the common factor.

$f' (x) = \frac{2 (- x^{2})}{2 {(30 - x^{2})}^{\frac{1}{2}}} + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.20.3

Rewrite the expression.

$f' (x) = \frac{- x^{2}}{{(30 - x^{2})}^{\frac{1}{2}}} + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.21

Move the negative in front of the fraction.

$f' (x) = - \frac{x^{2}}{{(30 - x^{2})}^{\frac{1}{2}}} + {(30 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.1.22

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

$f' (x) = - \frac{x^{2}}{{(30 - x^{2})}^{\frac{1}{2}}} + {(30 - x^{2})}^{\frac{1}{2}} \cdot 1$

Step 1.1.23

Multiply ${(30 - x^{2})}^{\frac{1}{2}}$ by $1$ .

$f' (x) = - \frac{x^{2}}{{(30 - x^{2})}^{\frac{1}{2}}} + {(30 - x^{2})}^{\frac{1}{2}}$

Step 1.1.24

To write ${(30 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(30 - x^{2})}^{\frac{1}{2}}}{{(30 - x^{2})}^{\frac{1}{2}}}$ .

$f' (x) = - \frac{x^{2}}{{(30 - x^{2})}^{\frac{1}{2}}} + \frac{{(30 - x^{2})}^{\frac{1}{2}} {(30 - x^{2})}^{\frac{1}{2}}}{{(30 - x^{2})}^{\frac{1}{2}}}$

Step 1.1.25

Combine the numerators over the common denominator.

$f' (x) = \frac{- x^{2} + {(30 - x^{2})}^{\frac{1}{2}} {(30 - x^{2})}^{\frac{1}{2}}}{{(30 - x^{2})}^{\frac{1}{2}}}$

Step 1.1.26

Multiply ${(30 - x^{2})}^{\frac{1}{2}}$ by ${(30 - x^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.1.26.1

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

$f' (x) = \frac{- x^{2} + {(30 - x^{2})}^{\frac{1}{2} + \frac{1}{2}}}{{(30 - x^{2})}^{\frac{1}{2}}}$

Step 1.1.26.2

Combine the numerators over the common denominator.

$f' (x) = \frac{- x^{2} + {(30 - x^{2})}^{\frac{1 + 1}{2}}}{{(30 - x^{2})}^{\frac{1}{2}}}$

Step 1.1.26.3

Add $1$ and $1$ .

$f' (x) = \frac{- x^{2} + {(30 - x^{2})}^{\frac{2}{2}}}{{(30 - x^{2})}^{\frac{1}{2}}}$

Step 1.1.26.4

Divide $2$ by $2$ .

$f' (x) = \frac{- x^{2} + 30 - x^{2}}{{(30 - x^{2})}^{\frac{1}{2}}}$

Step 1.1.27

Simplify ${(30 - x^{2})}^{1}$ .

$f' (x) = \frac{- x^{2} + 30 - x^{2}}{{(30 - x^{2})}^{\frac{1}{2}}}$

Step 1.1.28

Subtract $x^{2}$ from $- x^{2}$ .

$f' (x) = \frac{- 2 x^{2} + 30}{{(30 - x^{2})}^{\frac{1}{2}}}$

Step 1.1.29

Reorder terms.

$f' (x) = \frac{- 2 x^{2} + 30}{{(- x^{2} + 30)}^{\frac{1}{2}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{- 2 x^{2} + 30}{{(- x^{2} + 30)}^{\frac{1}{2}}}$ .

$\frac{- 2 x^{2} + 30}{{(- x^{2} + 30)}^{\frac{1}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{- 2 x^{2} + 30}{{(- x^{2} + 30)}^{\frac{1}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{- 2 x^{2} + 30}{{(- x^{2} + 30)}^{\frac{1}{2}}} = 0$

Step 2.2

Set the numerator equal to zero.

$- 2 x^{2} + 30 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Subtract $30$ from both sides of the equation.

$- 2 x^{2} = - 30$

Step 2.3.2

Divide each term in $- 2 x^{2} = - 30$ by $- 2$ and simplify.

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Step 2.3.2.1

Divide each term in $- 2 x^{2} = - 30$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{- 30}{- 2}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{- 30}{- 2}$

Step 2.3.2.2.1.2

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

$x^{2} = \frac{- 30}{- 2}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $- 30$ by $- 2$ .

$x^{2} = 15$

Step 2.3.3

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

$x = \pm \sqrt{15}$

Step 2.3.4

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

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Step 2.3.4.1

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

$x = \sqrt{15}$

Step 2.3.4.2

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

$x = - \sqrt{15}$

Step 2.3.4.3

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

$x = \sqrt{15}, - \sqrt{15}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{- 2 x^{2} + 30}{\sqrt{{(- x^{2} + 30)}^{1}}}$

Step 3.1.2

Anything raised to $1$ is the base itself.

$\frac{- 2 x^{2} + 30}{\sqrt{- x^{2} + 30}}$

Step 3.2

Set the denominator in $\frac{- 2 x^{2} + 30}{\sqrt{- x^{2} + 30}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{- x^{2} + 30} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{- x^{2} + 30}}^{2} = 0^{2}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- x^{2} + 30}$ as ${(- x^{2} + 30)}^{\frac{1}{2}}$ .

${({(- x^{2} + 30)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${({(- x^{2} + 30)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.2.1.1

Multiply the exponents in ${({(- x^{2} + 30)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.2.1.1.1

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

${(- x^{2} + 30)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.1.2.1

Cancel the common factor.

${(- x^{2} + 30)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.1.2.2

Rewrite the expression.

${(- x^{2} + 30)}^{1} = 0^{2}$

Step 3.3.2.2.1.2

Simplify.

$- x^{2} + 30 = 0^{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

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

$- x^{2} + 30 = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Subtract $30$ from both sides of the equation.

$- x^{2} = - 30$

Step 3.3.3.2

Divide each term in $- x^{2} = - 30$ by $- 1$ and simplify.

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Step 3.3.3.2.1

Divide each term in $- x^{2} = - 30$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 30}{- 1}$

Step 3.3.3.2.2

Simplify the left side.

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Step 3.3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 30}{- 1}$

Step 3.3.3.2.2.2

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

$x^{2} = \frac{- 30}{- 1}$

Step 3.3.3.2.3

Simplify the right side.

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Step 3.3.3.2.3.1

Divide $- 30$ by $- 1$ .

$x^{2} = 30$

Step 3.3.3.3

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

$x = \pm \sqrt{30}$

Step 3.3.3.4

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

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Step 3.3.3.4.1

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

$x = \sqrt{30}$

Step 3.3.3.4.2

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

$x = - \sqrt{30}$

Step 3.3.3.4.3

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

$x = \sqrt{30}, - \sqrt{30}$

Step 3.4

Set the radicand in $\sqrt{- x^{2} + 30}$ less than $0$ to find where the expression is undefined.

$- x^{2} + 30 < 0$

Step 3.5

Solve for $x$ .

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Step 3.5.1

Subtract $30$ from both sides of the inequality.

$- x^{2} < - 30$

Step 3.5.2

Divide each term in $- x^{2} < - 30$ by $- 1$ and simplify.

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Step 3.5.2.1

Divide each term in $- x^{2} < - 30$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 30}{- 1}$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} > \frac{- 30}{- 1}$

Step 3.5.2.2.2

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

$x^{2} > \frac{- 30}{- 1}$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Divide $- 30$ by $- 1$ .

$x^{2} > 30$

Step 3.5.3

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

$\sqrt{x^{2}} > \sqrt{30}$

Step 3.5.4

Simplify the left side.

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Step 3.5.4.1

Pull terms out from under the radical.

$| x | > \sqrt{30}$

Step 3.5.5

Write $| x | > \sqrt{30}$ as a piecewise.

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Step 3.5.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 3.5.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x > \sqrt{30}$

Step 3.5.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 3.5.5.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x > \sqrt{30}$

Step 3.5.5.5

Write as a piecewise.

${\begin{cases} x > \sqrt{30} & x \geq 0 \\ - x > \sqrt{30} & x < 0 \end{cases}$

Step 3.5.6

Find the intersection of $x > \sqrt{30}$ and $x \geq 0$ .

$x > \sqrt{30}$

Step 3.5.7

Divide each term in $- x > \sqrt{30}$ by $- 1$ and simplify.

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Step 3.5.7.1

Divide each term in $- x > \sqrt{30}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{\sqrt{30}}{- 1}$

Step 3.5.7.2

Simplify the left side.

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Step 3.5.7.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{\sqrt{30}}{- 1}$

Step 3.5.7.2.2

Divide $x$ by $1$ .

$x < \frac{\sqrt{30}}{- 1}$

Step 3.5.7.3

Simplify the right side.

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Step 3.5.7.3.1

Move the negative one from the denominator of $\frac{\sqrt{30}}{- 1}$ .

$x < - 1 \cdot \sqrt{30}$

Step 3.5.7.3.2

Rewrite $- 1 \cdot \sqrt{30}$ as $- \sqrt{30}$ .

$x < - \sqrt{30}$

Step 3.5.8

Find the union of the solutions.

$x < - \sqrt{30}$ or $x > \sqrt{30}$

Step 3.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq - \sqrt{30}, x \geq \sqrt{30}$

$(- \infty, - \sqrt{30}] \cup [\sqrt{30}, \infty)$

$x \leq - \sqrt{30}, x \geq \sqrt{30}$

$(- \infty, - \sqrt{30}] \cup [\sqrt{30}, \infty)$

Step 4

Evaluate $x \sqrt{30 - x^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \sqrt{15}$ .

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Step 4.1.1

Substitute $\sqrt{15}$ for $x$ .

$(\sqrt{15}) \sqrt{30 - {(\sqrt{15})}^{2}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Combine using the product rule for radicals.

$\sqrt{15 (30 - {(\sqrt{15})}^{2})}$

Step 4.1.2.2

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 4.1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\sqrt{15 (30 - {(15^{\frac{1}{2}})}^{2})}$

Step 4.1.2.2.2

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

$\sqrt{15 (30 - 15^{\frac{1}{2} \cdot 2})}$

Step 4.1.2.2.3

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

$\sqrt{15 (30 - 15^{\frac{2}{2}})}$

Step 4.1.2.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.2.4.1

Cancel the common factor.

$\sqrt{15 (30 - 15^{\frac{2}{2}})}$

Step 4.1.2.2.4.2

Rewrite the expression.

$\sqrt{15 (30 - 15^{1})}$

Step 4.1.2.2.5

Evaluate the exponent.

$\sqrt{15 (30 - 1 \cdot 15)}$

Step 4.1.2.3

Simplify the expression.

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Step 4.1.2.3.1

Multiply $- 1$ by $15$ .

$\sqrt{15 (30 - 15)}$

Step 4.1.2.3.2

Subtract $15$ from $30$ .

$\sqrt{15 \cdot 15}$

Step 4.1.2.3.3

Multiply $15$ by $15$ .

$\sqrt{225}$

Step 4.1.2.3.4

Rewrite $225$ as $15^{2}$ .

$\sqrt{15^{2}}$

Step 4.1.2.3.5

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

$15$

Step 4.2

Evaluate at $x = - \sqrt{15}$ .

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Step 4.2.1

Substitute $- \sqrt{15}$ for $x$ .

$(- \sqrt{15}) \sqrt{30 - {(- \sqrt{15})}^{2}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Apply the product rule to $- \sqrt{15}$ .

$- \sqrt{15} \sqrt{30 - ({(- 1)}^{2} {\sqrt{15}}^{2})}$

Step 4.2.2.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 4.2.2.2.1

Move ${(- 1)}^{2}$ .

$- \sqrt{15} \sqrt{30 + {(- 1)}^{2} \cdot - 1 {\sqrt{15}}^{2}}$

Step 4.2.2.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 4.2.2.2.2.1

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

$- \sqrt{15} \sqrt{30 + {(- 1)}^{2} \cdot {(- 1)}^{1} {\sqrt{15}}^{2}}$

Step 4.2.2.2.2.2

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

$- \sqrt{15} \sqrt{30 + {(- 1)}^{2 + 1} {\sqrt{15}}^{2}}$

Step 4.2.2.2.3

Add $2$ and $1$ .

$- \sqrt{15} \sqrt{30 + {(- 1)}^{3} {\sqrt{15}}^{2}}$

Step 4.2.2.3

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

$- \sqrt{15} \sqrt{30 - {\sqrt{15}}^{2}}$

Step 4.2.2.4

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 4.2.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$- \sqrt{15} \sqrt{30 - {(15^{\frac{1}{2}})}^{2}}$

Step 4.2.2.4.2

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

$- \sqrt{15} \sqrt{30 - 15^{\frac{1}{2} \cdot 2}}$

Step 4.2.2.4.3

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

$- \sqrt{15} \sqrt{30 - 15^{\frac{2}{2}}}$

Step 4.2.2.4.4

Cancel the common factor of $2$ .

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Step 4.2.2.4.4.1

Cancel the common factor.

$- \sqrt{15} \sqrt{30 - 15^{\frac{2}{2}}}$

Step 4.2.2.4.4.2

Rewrite the expression.

$- \sqrt{15} \sqrt{30 - 15^{1}}$

Step 4.2.2.4.5

Evaluate the exponent.

$- \sqrt{15} \sqrt{30 - 1 \cdot 15}$

Step 4.2.2.5

Simplify the expression.

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Step 4.2.2.5.1

Multiply $- 1$ by $15$ .

$- \sqrt{15} \sqrt{30 - 15}$

Step 4.2.2.5.2

Subtract $15$ from $30$ .

$- \sqrt{15} \sqrt{15}$

Step 4.2.2.6

Multiply $- \sqrt{15} \sqrt{15}$ .

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Step 4.2.2.6.1

Raise $\sqrt{15}$ to the power of $1$ .

$- ({\sqrt{15}}^{1} \sqrt{15})$

Step 4.2.2.6.2

Raise $\sqrt{15}$ to the power of $1$ .

$- ({\sqrt{15}}^{1} {\sqrt{15}}^{1})$

Step 4.2.2.6.3

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

$- {\sqrt{15}}^{1 + 1}$

Step 4.2.2.6.4

Add $1$ and $1$ .

$- {\sqrt{15}}^{2}$

Step 4.2.2.7

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 4.2.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$- {(15^{\frac{1}{2}})}^{2}$

Step 4.2.2.7.2

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

$- 15^{\frac{1}{2} \cdot 2}$

Step 4.2.2.7.3

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

$- 15^{\frac{2}{2}}$

Step 4.2.2.7.4

Cancel the common factor of $2$ .

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Step 4.2.2.7.4.1

Cancel the common factor.

$- 15^{\frac{2}{2}}$

Step 4.2.2.7.4.2

Rewrite the expression.

$- 15^{1}$

Step 4.2.2.7.5

Evaluate the exponent.

$- 1 \cdot 15$

Step 4.2.2.8

Multiply $- 1$ by $15$ .

$- 15$

Step 4.3

Evaluate at $x = - \sqrt{30}$ .

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Step 4.3.1

Substitute $- \sqrt{30}$ for $x$ .

$(- \sqrt{30}) \sqrt{30 - {(- \sqrt{30})}^{2}}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Apply the product rule to $- \sqrt{30}$ .

$- \sqrt{30} \sqrt{30 - ({(- 1)}^{2} {\sqrt{30}}^{2})}$

Step 4.3.2.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 4.3.2.2.1

Move ${(- 1)}^{2}$ .

$- \sqrt{30} \sqrt{30 + {(- 1)}^{2} \cdot - 1 {\sqrt{30}}^{2}}$

Step 4.3.2.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 4.3.2.2.2.1

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

$- \sqrt{30} \sqrt{30 + {(- 1)}^{2} \cdot {(- 1)}^{1} {\sqrt{30}}^{2}}$

Step 4.3.2.2.2.2

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

$- \sqrt{30} \sqrt{30 + {(- 1)}^{2 + 1} {\sqrt{30}}^{2}}$

Step 4.3.2.2.3

Add $2$ and $1$ .

$- \sqrt{30} \sqrt{30 + {(- 1)}^{3} {\sqrt{30}}^{2}}$

Step 4.3.2.3

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

$- \sqrt{30} \sqrt{30 - {\sqrt{30}}^{2}}$

Step 4.3.2.4

Rewrite ${\sqrt{30}}^{2}$ as $30$ .

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Step 4.3.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{30}$ as $30^{\frac{1}{2}}$ .

$- \sqrt{30} \sqrt{30 - {(30^{\frac{1}{2}})}^{2}}$

Step 4.3.2.4.2

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

$- \sqrt{30} \sqrt{30 - 30^{\frac{1}{2} \cdot 2}}$

Step 4.3.2.4.3

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

$- \sqrt{30} \sqrt{30 - 30^{\frac{2}{2}}}$

Step 4.3.2.4.4

Cancel the common factor of $2$ .

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Step 4.3.2.4.4.1

Cancel the common factor.

$- \sqrt{30} \sqrt{30 - 30^{\frac{2}{2}}}$

Step 4.3.2.4.4.2

Rewrite the expression.

$- \sqrt{30} \sqrt{30 - 30^{1}}$

Step 4.3.2.4.5

Evaluate the exponent.

$- \sqrt{30} \sqrt{30 - 1 \cdot 30}$

Step 4.3.2.5

Simplify the expression.

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Step 4.3.2.5.1

Multiply $- 1$ by $30$ .

$- \sqrt{30} \sqrt{30 - 30}$

Step 4.3.2.5.2

Subtract $30$ from $30$ .

$- \sqrt{30} \sqrt{0}$

Step 4.3.2.5.3

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

$- \sqrt{30} \sqrt{0^{2}}$

Step 4.3.2.5.4

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

$- \sqrt{30} \cdot 0$

Step 4.3.2.6

Multiply $- \sqrt{30} \cdot 0$ .

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Step 4.3.2.6.1

Multiply $0$ by $- 1$ .

$0 \sqrt{30}$

Step 4.3.2.6.2

Multiply $0$ by $\sqrt{30}$ .

$0$

Step 4.4

Evaluate at $x = \sqrt{30}$ .

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Step 4.4.1

Substitute $\sqrt{30}$ for $x$ .

$(\sqrt{30}) \sqrt{30 - {(\sqrt{30})}^{2}}$

Step 4.4.2

Simplify.

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Step 4.4.2.1

Combine using the product rule for radicals.

$\sqrt{30 (30 - {(\sqrt{30})}^{2})}$

Step 4.4.2.2

Rewrite ${\sqrt{30}}^{2}$ as $30$ .

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Step 4.4.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{30}$ as $30^{\frac{1}{2}}$ .

$\sqrt{30 (30 - {(30^{\frac{1}{2}})}^{2})}$

Step 4.4.2.2.2

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

$\sqrt{30 (30 - 30^{\frac{1}{2} \cdot 2})}$

Step 4.4.2.2.3

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

$\sqrt{30 (30 - 30^{\frac{2}{2}})}$

Step 4.4.2.2.4

Cancel the common factor of $2$ .

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Step 4.4.2.2.4.1

Cancel the common factor.

$\sqrt{30 (30 - 30^{\frac{2}{2}})}$

Step 4.4.2.2.4.2

Rewrite the expression.

$\sqrt{30 (30 - 30^{1})}$

Step 4.4.2.2.5

Evaluate the exponent.

$\sqrt{30 (30 - 1 \cdot 30)}$

Step 4.4.2.3

Simplify the expression.

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Step 4.4.2.3.1

Multiply $- 1$ by $30$ .

$\sqrt{30 (30 - 30)}$

Step 4.4.2.3.2

Subtract $30$ from $30$ .

$\sqrt{30 \cdot 0}$

Step 4.4.2.3.3

Multiply $30$ by $0$ .

$\sqrt{0}$

Step 4.4.2.3.4

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

$\sqrt{0^{2}}$

Step 4.4.2.3.5

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

$0$

Step 4.5

List all of the points.

$(\sqrt{15}, 15), (- \sqrt{15}, - 15), (- \sqrt{30}, 0), (\sqrt{30}, 0)$

Step 5