Calculus Examples

$f (x) = 188 {(\frac{200}{x} + \frac{x}{15})}^{- 1}$

Step 1

Find the first derivative of the function.

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Step 1.1

Since $188$ is constant with respect to $x$ , the derivative of $188 {(\frac{200}{x} + \frac{x}{15})}^{- 1}$ with respect to $x$ is $188 \frac{d}{d x} [{(\frac{200}{x} + \frac{x}{15})}^{- 1}]$ .

$188 \frac{d}{d x} [{(\frac{200}{x} + \frac{x}{15})}^{- 1}]$

Step 1.2

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 $g (x) = \frac{200}{x} + \frac{x}{15}$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $\frac{200}{x} + \frac{x}{15}$ .

$188 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [\frac{200}{x} + \frac{x}{15}])$

Step 1.2.2

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

$188 (- u^{- 2} \frac{d}{d x} [\frac{200}{x} + \frac{x}{15}])$

Step 1.2.3

Replace all occurrences of $u$ with $\frac{200}{x} + \frac{x}{15}$ .

$188 (- {(\frac{200}{x} + \frac{x}{15})}^{- 2} \frac{d}{d x} [\frac{200}{x} + \frac{x}{15}])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $188$ .

$- 188 ({(\frac{200}{x} + \frac{x}{15})}^{- 2} \frac{d}{d x} [\frac{200}{x} + \frac{x}{15}])$

Step 1.3.2

By the Sum Rule, the derivative of $\frac{200}{x} + \frac{x}{15}$ with respect to $x$ is $\frac{d}{d x} [\frac{200}{x}] + \frac{d}{d x} [\frac{x}{15}]$ .

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (\frac{d}{d x} [\frac{200}{x}] + \frac{d}{d x} [\frac{x}{15}])$

Step 1.3.3

Since $200$ is constant with respect to $x$ , the derivative of $\frac{200}{x}$ with respect to $x$ is $200 \frac{d}{d x} [\frac{1}{x}]$ .

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (200 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\frac{x}{15}])$

Step 1.3.4

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

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (200 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [\frac{x}{15}])$

Step 1.3.5

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

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (200 (- x^{- 2}) + \frac{d}{d x} [\frac{x}{15}])$

Step 1.3.6

Multiply $- 1$ by $200$ .

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{d}{d x} [\frac{x}{15}])$

Step 1.3.7

Since $\frac{1}{15}$ is constant with respect to $x$ , the derivative of $\frac{x}{15}$ with respect to $x$ is $\frac{1}{15} \frac{d}{d x} [x]$ .

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15} \frac{d}{d x} [x])$

Step 1.3.8

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

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15} \cdot 1)$

Step 1.3.9

Multiply $\frac{1}{15}$ by $1$ .

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15})$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- 188$ is constant with respect to $x$ , the derivative of $- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15})$ with respect to $x$ is $- 188 \frac{d}{d x} [{(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15})]$ .

$f'' (x) = - 188 \frac{d}{d x} ({(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15}))$

Step 2.2

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 $f (x) = {(\frac{200}{x} + \frac{x}{15})}^{- 2}$ and $g (x) = - 200 x^{- 2} + \frac{1}{15}$ .

$f'' (x) = - 188 ({(\frac{200}{x} + \frac{x}{15})}^{- 2} \frac{d}{d x} (- 200 x^{- 2} + \frac{1}{15}) + (- 200 x^{- 2} + \frac{1}{15}) \frac{d}{d x} ({(\frac{200}{x} + \frac{x}{15})}^{- 2}))$

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $- 200 x^{- 2} + \frac{1}{15}$ with respect to $x$ is $\frac{d}{d x} [- 200 x^{- 2}] + \frac{d}{d x} [\frac{1}{15}]$ .

$f'' (x) = - 188 ({(\frac{200}{x} + \frac{x}{15})}^{- 2} (\frac{d}{d x} (- 200 x^{- 2}) + \frac{d}{d x} (\frac{1}{15})) + (- 200 x^{- 2} + \frac{1}{15}) \frac{d}{d x} ({(\frac{200}{x} + \frac{x}{15})}^{- 2}))$

Step 2.3.2

Since $- 200$ is constant with respect to $x$ , the derivative of $- 200 x^{- 2}$ with respect to $x$ is $- 200 \frac{d}{d x} [x^{- 2}]$ .

$f'' (x) = - 188 ({(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 \frac{d}{d x} x^{- 2} + \frac{d}{d x} (\frac{1}{15})) + (- 200 x^{- 2} + \frac{1}{15}) \frac{d}{d x} ({(\frac{200}{x} + \frac{x}{15})}^{- 2}))$

Step 2.3.3

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

$f'' (x) = - 188 ({(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 (- 2 x^{- 3}) + \frac{d}{d x} (\frac{1}{15})) + (- 200 x^{- 2} + \frac{1}{15}) \frac{d}{d x} ({(\frac{200}{x} + \frac{x}{15})}^{- 2}))$

Step 2.3.4

Multiply $- 2$ by $- 200$ .

$f'' (x) = - 188 ({(\frac{200}{x} + \frac{x}{15})}^{- 2} (400 x^{- 3} + \frac{d}{d x} (\frac{1}{15})) + (- 200 x^{- 2} + \frac{1}{15}) \frac{d}{d x} ({(\frac{200}{x} + \frac{x}{15})}^{- 2}))$

Step 2.3.5

Since $\frac{1}{15}$ is constant with respect to $x$ , the derivative of $\frac{1}{15}$ with respect to $x$ is $0$ .

$f'' (x) = - 188 ({(\frac{200}{x} + \frac{x}{15})}^{- 2} (400 x^{- 3} + 0) + (- 200 x^{- 2} + \frac{1}{15}) \frac{d}{d x} ({(\frac{200}{x} + \frac{x}{15})}^{- 2}))$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $400 x^{- 3}$ and $0$ .

$f'' (x) = - 188 ({(\frac{200}{x} + \frac{x}{15})}^{- 2} (400 x^{- 3}) + (- 200 x^{- 2} + \frac{1}{15}) \frac{d}{d x} ({(\frac{200}{x} + \frac{x}{15})}^{- 2}))$

Step 2.3.6.2

Move $400$ to the left of ${(\frac{200}{x} + \frac{x}{15})}^{- 2}$ .

$f'' (x) = - 188 (400 \cdot ({(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3}) + (- 200 x^{- 2} + \frac{1}{15}) \frac{d}{d x} ({(\frac{200}{x} + \frac{x}{15})}^{- 2}))$

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 2}$ and $g (x) = \frac{200}{x} + \frac{x}{15}$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $\frac{200}{x} + \frac{x}{15}$ .

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} + (- 200 x^{- 2} + \frac{1}{15}) (\frac{d}{d u} (u^{- 2}) \frac{d}{d x} (\frac{200}{x} + \frac{x}{15})))$

Step 2.4.2

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

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} + (- 200 x^{- 2} + \frac{1}{15}) (- 2 u^{- 3} \frac{d}{d x} (\frac{200}{x} + \frac{x}{15})))$

Step 2.4.3

Replace all occurrences of $u$ with $\frac{200}{x} + \frac{x}{15}$ .

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} + (- 200 x^{- 2} + \frac{1}{15}) (- 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} \frac{d}{d x} (\frac{200}{x} + \frac{x}{15})))$

Step 2.5

Differentiate.

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Step 2.5.1

By the Sum Rule, the derivative of $\frac{200}{x} + \frac{x}{15}$ with respect to $x$ is $\frac{d}{d x} [\frac{200}{x}] + \frac{d}{d x} [\frac{x}{15}]$ .

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} + (- 200 x^{- 2} + \frac{1}{15}) (- 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} (\frac{d}{d x} (\frac{200}{x}) + \frac{d}{d x} (\frac{x}{15}))))$

Step 2.5.2

Since $200$ is constant with respect to $x$ , the derivative of $\frac{200}{x}$ with respect to $x$ is $200 \frac{d}{d x} [\frac{1}{x}]$ .

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} + (- 200 x^{- 2} + \frac{1}{15}) (- 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} (200 \frac{d}{d x} (\frac{1}{x}) + \frac{d}{d x} (\frac{x}{15}))))$

Step 2.5.3

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

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} + (- 200 x^{- 2} + \frac{1}{15}) (- 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} (200 \frac{d}{d x} (x^{- 1}) + \frac{d}{d x} (\frac{x}{15}))))$

Step 2.5.4

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

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} + (- 200 x^{- 2} + \frac{1}{15}) (- 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} (200 (- x^{- 2}) + \frac{d}{d x} (\frac{x}{15}))))$

Step 2.5.5

Multiply $- 1$ by $200$ .

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} + (- 200 x^{- 2} + \frac{1}{15}) (- 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} (- 200 x^{- 2} + \frac{d}{d x} (\frac{x}{15}))))$

Step 2.5.6

Since $\frac{1}{15}$ is constant with respect to $x$ , the derivative of $\frac{x}{15}$ with respect to $x$ is $\frac{1}{15} \frac{d}{d x} [x]$ .

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} + (- 200 x^{- 2} + \frac{1}{15}) (- 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} (- 200 x^{- 2} + \frac{1}{15} \cdot \frac{d}{d x} (x))))$

Step 2.5.7

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

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} + (- 200 x^{- 2} + \frac{1}{15}) (- 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} (- 200 x^{- 2} + \frac{1}{15} \cdot 1)))$

Step 2.5.8

Multiply $\frac{1}{15}$ by $1$ .

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} + (- 200 x^{- 2} + \frac{1}{15}) (- 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} (- 200 x^{- 2} + \frac{1}{15})))$

Step 2.6

Raise $- 200 x^{- 2} + \frac{1}{15}$ to the power of $1$ .

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} - 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} ((- 200 x^{- 2} + \frac{1}{15}) (- 200 x^{- 2} + \frac{1}{15})))$

Step 2.7

Raise $- 200 x^{- 2} + \frac{1}{15}$ to the power of $1$ .

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} - 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} ((- 200 x^{- 2} + \frac{1}{15}) (- 200 x^{- 2} + \frac{1}{15})))$

Step 2.8

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

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} - 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} {(- 200 x^{- 2} + \frac{1}{15})}^{1 + 1})$

Step 2.9

Add $1$ and $1$ .

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} - 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} {(- 200 x^{- 2} + \frac{1}{15})}^{2})$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$f'' (x) = - 188 (400 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3}) - 188 (- 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} {(- 200 x^{- 2} + \frac{1}{15})}^{2})$

Step 2.10.2

Combine terms.

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Step 2.10.2.1

Multiply $400$ by $- 188$ .

$f'' (x) = - 75200 ({(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3}) - 188 (- 2 {(\frac{200}{x} + \frac{x}{15})}^{- 3} {(- 200 x^{- 2} + \frac{1}{15})}^{2})$

Step 2.10.2.2

Multiply $- 2$ by $- 188$ .

$f'' (x) = - 75200 {(\frac{200}{x} + \frac{x}{15})}^{- 2} x^{- 3} + 376 {(\frac{200}{x} + \frac{x}{15})}^{- 3} {(- 200 x^{- 2} + \frac{1}{15})}^{2}$

Step 2.10.3

Reorder terms.

$f'' (x) = - 75200 x^{- 3} {(\frac{200}{x} + \frac{x}{15})}^{- 2} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4

Simplify each term.

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Step 2.10.4.1

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

$f'' (x) = - 75200 \frac{1}{x^{3}} \cdot {(\frac{200}{x} + \frac{x}{15})}^{- 2} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.2

Combine $- 75200$ and $\frac{1}{x^{3}}$ .

$f'' (x) = \frac{- 75200}{x^{3}} \cdot {(\frac{200}{x} + \frac{x}{15})}^{- 2} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.3

Move the negative in front of the fraction.

$f'' (x) = - \frac{75200}{x^{3}} \cdot {(\frac{200}{x} + \frac{x}{15})}^{- 2} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.4

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

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{1}{{(\frac{200}{x} + \frac{x}{15})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.5

Simplify the denominator.

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Step 2.10.4.5.1

To write $\frac{200}{x}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{1}{{(\frac{200}{x} \cdot \frac{15}{15} + \frac{x}{15})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.5.2

To write $\frac{x}{15}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{1}{{(\frac{200}{x} \cdot \frac{15}{15} + \frac{x}{15} \cdot \frac{x}{x})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.5.3

Write each expression with a common denominator of $x \cdot 15$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.10.4.5.3.1

Multiply $\frac{200}{x}$ by $\frac{15}{15}$ .

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{1}{{(\frac{200 \cdot 15}{x \cdot 15} + \frac{x}{15} \cdot \frac{x}{x})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.5.3.2

Multiply $\frac{x}{15}$ by $\frac{x}{x}$ .

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{1}{{(\frac{200 \cdot 15}{x \cdot 15} + \frac{x \cdot x}{15 x})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.5.3.3

Reorder the factors of $x \cdot 15$ .

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{1}{{(\frac{200 \cdot 15}{15 x} + \frac{x \cdot x}{15 x})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.5.4

Combine the numerators over the common denominator.

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{1}{{(\frac{200 \cdot 15 + x \cdot x}{15 x})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.5.5

Simplify the numerator.

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Step 2.10.4.5.5.1

Multiply $200$ by $15$ .

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{1}{{(\frac{3000 + x \cdot x}{15 x})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.5.5.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{1}{{(\frac{3000 + x^{2}}{15 x})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.5.6

Apply the product rule to $\frac{3000 + x^{2}}{15 x}$ .

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{1}{\frac{{(3000 + x^{2})}^{2}}{{(15 x)}^{2}}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.5.7

Apply the product rule to $15 x$ .

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{1}{\frac{{(3000 + x^{2})}^{2}}{15^{2} x^{2}}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.5.8

Raise $15$ to the power of $2$ .

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{1}{\frac{{(3000 + x^{2})}^{2}}{225 x^{2}}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.6

Multiply the numerator by the reciprocal of the denominator.

$f'' (x) = - \frac{75200}{x^{3}} \cdot (1 (\frac{225 x^{2}}{{(3000 + x^{2})}^{2}})) + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.7

Multiply $\frac{225 x^{2}}{{(3000 + x^{2})}^{2}}$ by $1$ .

$f'' (x) = - \frac{75200}{x^{3}} \cdot \frac{225 x^{2}}{{(3000 + x^{2})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.8

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

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Step 2.10.4.8.1

Move the leading negative in $- \frac{75200}{x^{3}}$ into the numerator.

$f'' (x) = \frac{- 75200}{x^{3}} \cdot \frac{225 x^{2}}{{(3000 + x^{2})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.8.2

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

$f'' (x) = \frac{- 75200}{x^{2} x} \cdot \frac{225 x^{2}}{{(3000 + x^{2})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.8.3

Factor $x^{2}$ out of $225 x^{2}$ .

$f'' (x) = \frac{- 75200}{x^{2} x} \cdot \frac{x^{2} \cdot 225}{{(3000 + x^{2})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.8.4

Cancel the common factor.

$f'' (x) = \frac{- 75200}{x^{2} x} \cdot \frac{x^{2} \cdot 225}{{(3000 + x^{2})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.8.5

Rewrite the expression.

$f'' (x) = \frac{- 75200}{x} \cdot \frac{225}{{(3000 + x^{2})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.9

Multiply $\frac{- 75200}{x}$ by $\frac{225}{{(3000 + x^{2})}^{2}}$ .

$f'' (x) = \frac{- 75200 \cdot 225}{x {(3000 + x^{2})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.10

Multiply $- 75200$ by $225$ .

$f'' (x) = \frac{- 16920000}{x {(3000 + x^{2})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.11

Move the negative in front of the fraction.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 {(- 200 x^{- 2} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.12

Simplify each term.

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Step 2.10.4.12.1

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

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 {(- 200 \frac{1}{x^{2}} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.12.2

Combine $- 200$ and $\frac{1}{x^{2}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 {(\frac{- 200}{x^{2}} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.12.3

Move the negative in front of the fraction.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 {(- \frac{200}{x^{2}} + \frac{1}{15})}^{2} {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.13

Rewrite ${(- \frac{200}{x^{2}} + \frac{1}{15})}^{2}$ as $(- \frac{200}{x^{2}} + \frac{1}{15}) (- \frac{200}{x^{2}} + \frac{1}{15})$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 ((- \frac{200}{x^{2}} + \frac{1}{15}) (- \frac{200}{x^{2}} + \frac{1}{15})) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.14

Expand $(- \frac{200}{x^{2}} + \frac{1}{15}) (- \frac{200}{x^{2}} + \frac{1}{15})$ using the FOIL Method.

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Step 2.10.4.14.1

Apply the distributive property.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (- \frac{200}{x^{2}} \cdot (- \frac{200}{x^{2}} + \frac{1}{15}) + \frac{1}{15} \cdot (- \frac{200}{x^{2}} + \frac{1}{15})) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.14.2

Apply the distributive property.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (- \frac{200}{x^{2}} \cdot (- \frac{200}{x^{2}}) - \frac{200}{x^{2}} \cdot \frac{1}{15} + \frac{1}{15} \cdot (- \frac{200}{x^{2}} + \frac{1}{15})) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.14.3

Apply the distributive property.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (- \frac{200}{x^{2}} \cdot (- \frac{200}{x^{2}}) - \frac{200}{x^{2}} \cdot \frac{1}{15} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15

Simplify and combine like terms.

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Step 2.10.4.15.1

Simplify each term.

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Step 2.10.4.15.1.1

Multiply $- \frac{200}{x^{2}} (- \frac{200}{x^{2}})$ .

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Step 2.10.4.15.1.1.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (1 (\frac{200}{x^{2}}) \cdot \frac{200}{x^{2}} - \frac{200}{x^{2}} \cdot \frac{1}{15} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.1.2

Multiply $\frac{200}{x^{2}}$ by $1$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{200}{x^{2}} \cdot \frac{200}{x^{2}} - \frac{200}{x^{2}} \cdot \frac{1}{15} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.1.3

Multiply $\frac{200}{x^{2}}$ by $\frac{200}{x^{2}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{200 \cdot 200}{x^{2} x^{2}} - \frac{200}{x^{2}} \cdot \frac{1}{15} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.1.4

Multiply $200$ by $200$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{2} x^{2}} - \frac{200}{x^{2}} \cdot \frac{1}{15} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.1.5

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

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Step 2.10.4.15.1.1.5.1

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

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{2 + 2}} - \frac{200}{x^{2}} \cdot \frac{1}{15} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.1.5.2

Add $2$ and $2$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} - \frac{200}{x^{2}} \cdot \frac{1}{15} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.2

Cancel the common factor of $5$ .

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Step 2.10.4.15.1.2.1

Move the leading negative in $- \frac{200}{x^{2}}$ into the numerator.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} + \frac{- 200}{x^{2}} \cdot \frac{1}{15} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.2.2

Factor $5$ out of $- 200$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} + \frac{5 (- 40)}{x^{2}} \cdot \frac{1}{15} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.2.3

Factor $5$ out of $15$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} + \frac{5 \cdot - 40}{x^{2}} \cdot \frac{1}{5 \cdot 3} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.2.4

Cancel the common factor.

Step 2.10.4.15.1.2.5

Rewrite the expression.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} + \frac{- 40}{x^{2}} \cdot \frac{1}{3} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.3

Multiply $\frac{- 40}{x^{2}}$ by $\frac{1}{3}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} + \frac{- 40}{x^{2} \cdot 3} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.4

Move $3$ to the left of $x^{2}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} + \frac{- 40}{3 \cdot x^{2}} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.5

Move the negative in front of the fraction.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} - \frac{40}{3 \cdot x^{2}} + \frac{1}{15} \cdot (- \frac{200}{x^{2}}) + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.6

Cancel the common factor of $5$ .

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Step 2.10.4.15.1.6.1

Move the leading negative in $- \frac{200}{x^{2}}$ into the numerator.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} - \frac{40}{3 x^{2}} + \frac{1}{15} \cdot \frac{- 200}{x^{2}} + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.6.2

Factor $5$ out of $15$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} - \frac{40}{3 x^{2}} + \frac{1}{5 (3)} \cdot \frac{- 200}{x^{2}} + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.6.3

Factor $5$ out of $- 200$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} - \frac{40}{3 x^{2}} + \frac{1}{5 \cdot 3} \cdot \frac{5 \cdot - 40}{x^{2}} + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.6.4

Cancel the common factor.

Step 2.10.4.15.1.6.5

Rewrite the expression.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} - \frac{40}{3 x^{2}} + \frac{1}{3} \cdot \frac{- 40}{x^{2}} + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.7

Multiply $\frac{1}{3}$ by $\frac{- 40}{x^{2}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} - \frac{40}{3 x^{2}} + \frac{- 40}{3 x^{2}} + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.8

Move the negative in front of the fraction.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} - \frac{40}{3 x^{2}} - \frac{40}{3 x^{2}} + \frac{1}{15} \cdot \frac{1}{15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.9

Multiply $\frac{1}{15} \cdot \frac{1}{15}$ .

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Step 2.10.4.15.1.9.1

Multiply $\frac{1}{15}$ by $\frac{1}{15}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} - \frac{40}{3 x^{2}} - \frac{40}{3 x^{2}} + \frac{1}{15 \cdot 15}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.1.9.2

Multiply $15$ by $15$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} - \frac{40}{3 x^{2}} - \frac{40}{3 x^{2}} + \frac{1}{225}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.15.2

Subtract $\frac{40}{3 x^{2}}$ from $- \frac{40}{3 x^{2}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} - 2 \frac{40}{3 x^{2}} + \frac{1}{225}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.16

Simplify each term.

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Step 2.10.4.16.1

Multiply $- 2 \frac{40}{3 x^{2}}$ .

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Step 2.10.4.16.1.1

Combine $- 2$ and $\frac{40}{3 x^{2}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} + \frac{- 2 \cdot 40}{3 x^{2}} + \frac{1}{225}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.16.1.2

Multiply $- 2$ by $40$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} + \frac{- 80}{3 x^{2}} + \frac{1}{225}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.16.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + 376 (\frac{40000}{x^{4}} - \frac{80}{3 x^{2}} + \frac{1}{225}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.17

Apply the distributive property.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (376 (\frac{40000}{x^{4}}) + 376 (- \frac{80}{3 x^{2}}) + 376 (\frac{1}{225})) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.18

Simplify.

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Step 2.10.4.18.1

Multiply $376 \frac{40000}{x^{4}}$ .

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Step 2.10.4.18.1.1

Combine $376$ and $\frac{40000}{x^{4}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{376 \cdot 40000}{x^{4}} + 376 (- \frac{80}{3 x^{2}}) + 376 (\frac{1}{225})) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.18.1.2

Multiply $376$ by $40000$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} + 376 (- \frac{80}{3 x^{2}}) + 376 (\frac{1}{225})) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.18.2

Multiply $376 (- \frac{80}{3 x^{2}})$ .

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Step 2.10.4.18.2.1

Multiply $- 1$ by $376$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - 376 \frac{80}{3 x^{2}} + 376 (\frac{1}{225})) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.18.2.2

Combine $- 376$ and $\frac{80}{3 x^{2}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} + \frac{- 376 \cdot 80}{3 x^{2}} + 376 (\frac{1}{225})) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.18.2.3

Multiply $- 376$ by $80$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} + \frac{- 30080}{3 x^{2}} + 376 (\frac{1}{225})) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.18.3

Combine $376$ and $\frac{1}{225}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} + \frac{- 30080}{3 x^{2}} + \frac{376}{225}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.19

Move the negative in front of the fraction.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) {(\frac{200}{x} + \frac{x}{15})}^{- 3}$

Step 2.10.4.20

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

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{1}{{(\frac{200}{x} + \frac{x}{15})}^{3}})$

Step 2.10.4.21

Simplify the denominator.

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Step 2.10.4.21.1

To write $\frac{200}{x}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{1}{{(\frac{200}{x} \cdot \frac{15}{15} + \frac{x}{15})}^{3}})$

Step 2.10.4.21.2

To write $\frac{x}{15}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{1}{{(\frac{200}{x} \cdot \frac{15}{15} + \frac{x}{15} \cdot \frac{x}{x})}^{3}})$

Step 2.10.4.21.3

Write each expression with a common denominator of $x \cdot 15$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.10.4.21.3.1

Multiply $\frac{200}{x}$ by $\frac{15}{15}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{1}{{(\frac{200 \cdot 15}{x \cdot 15} + \frac{x}{15} \cdot \frac{x}{x})}^{3}})$

Step 2.10.4.21.3.2

Multiply $\frac{x}{15}$ by $\frac{x}{x}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{1}{{(\frac{200 \cdot 15}{x \cdot 15} + \frac{x \cdot x}{15 x})}^{3}})$

Step 2.10.4.21.3.3

Reorder the factors of $x \cdot 15$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{1}{{(\frac{200 \cdot 15}{15 x} + \frac{x \cdot x}{15 x})}^{3}})$

Step 2.10.4.21.4

Combine the numerators over the common denominator.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{1}{{(\frac{200 \cdot 15 + x \cdot x}{15 x})}^{3}})$

Step 2.10.4.21.5

Simplify the numerator.

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Step 2.10.4.21.5.1

Multiply $200$ by $15$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{1}{{(\frac{3000 + x \cdot x}{15 x})}^{3}})$

Step 2.10.4.21.5.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{1}{{(\frac{3000 + x^{2}}{15 x})}^{3}})$

Step 2.10.4.21.6

Apply the product rule to $\frac{3000 + x^{2}}{15 x}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{1}{\frac{{(3000 + x^{2})}^{3}}{{(15 x)}^{3}}})$

Step 2.10.4.21.7

Apply the product rule to $15 x$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{1}{\frac{{(3000 + x^{2})}^{3}}{15^{3} x^{3}}})$

Step 2.10.4.21.8

Raise $15$ to the power of $3$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{1}{\frac{{(3000 + x^{2})}^{3}}{3375 x^{3}}})$

Step 2.10.4.22

Multiply the numerator by the reciprocal of the denominator.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (1 (\frac{3375 x^{3}}{{(3000 + x^{2})}^{3}}))$

Step 2.10.4.23

Multiply $\frac{3375 x^{3}}{{(3000 + x^{2})}^{3}}$ by $1$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + (\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (\frac{3375 x^{3}}{{(3000 + x^{2})}^{3}})$

Step 2.10.4.24

Multiply $\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}$ by $\frac{3375 x^{3}}{{(3000 + x^{2})}^{3}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(\frac{15040000}{x^{4}} - \frac{30080}{3 x^{2}} + \frac{376}{225}) (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25

Simplify the numerator.

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Step 2.10.4.25.1

Reorder terms.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(- \frac{30080}{3 x^{2}} + \frac{15040000}{x^{4}} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.2

To write $- \frac{30080}{3 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(- \frac{30080}{3 x^{2}} \cdot \frac{x^{2}}{x^{2}} + \frac{15040000}{x^{4}} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.3

To write $\frac{15040000}{x^{4}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(- \frac{30080}{3 x^{2}} \cdot \frac{x^{2}}{x^{2}} + \frac{15040000}{x^{4}} \cdot \frac{3}{3} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.4

Write each expression with a common denominator of $3 x^{4}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.10.4.25.4.1

Multiply $\frac{30080}{3 x^{2}}$ by $\frac{x^{2}}{x^{2}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(- \frac{30080 x^{2}}{3 x^{2} x^{2}} + \frac{15040000}{x^{4}} \cdot \frac{3}{3} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.4.2

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

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Step 2.10.4.25.4.2.1

Move $x^{2}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(- \frac{30080 x^{2}}{3 (x^{2} x^{2})} + \frac{15040000}{x^{4}} \cdot \frac{3}{3} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.4.2.2

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

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(- \frac{30080 x^{2}}{3 x^{2 + 2}} + \frac{15040000}{x^{4}} \cdot \frac{3}{3} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.4.2.3

Add $2$ and $2$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(- \frac{30080 x^{2}}{3 x^{4}} + \frac{15040000}{x^{4}} \cdot \frac{3}{3} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.4.3

Multiply $\frac{15040000}{x^{4}}$ by $\frac{3}{3}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(- \frac{30080 x^{2}}{3 x^{4}} + \frac{15040000 \cdot 3}{x^{4} \cdot 3} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.4.4

Reorder the factors of $x^{4} \cdot 3$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(- \frac{30080 x^{2}}{3 x^{4}} + \frac{15040000 \cdot 3}{3 x^{4}} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.5

Combine the numerators over the common denominator.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(\frac{- 30080 x^{2} + 15040000 \cdot 3}{3 x^{4}} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.6

Multiply $15040000$ by $3$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(\frac{- 30080 x^{2} + 45120000}{3 x^{4}} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.7

Factor $30080$ out of $- 30080 x^{2} + 45120000$ .

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Step 2.10.4.25.7.1

Factor $30080$ out of $- 30080 x^{2}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(\frac{30080 (- x^{2}) + 45120000}{3 x^{4}} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.7.2

Factor $30080$ out of $45120000$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(\frac{30080 (- x^{2}) + 30080 (1500)}{3 x^{4}} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.7.3

Factor $30080$ out of $30080 (- x^{2}) + 30080 (1500)$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(\frac{30080 (- x^{2} + 1500)}{3 x^{4}} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.8

To write $\frac{30080 (- x^{2} + 1500)}{3 x^{4}}$ as a fraction with a common denominator, multiply by $\frac{75}{75}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(\frac{30080 (- x^{2} + 1500)}{3 x^{4}} \cdot \frac{75}{75} + \frac{376}{225}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.9

To write $\frac{376}{225}$ as a fraction with a common denominator, multiply by $\frac{x^{4}}{x^{4}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(\frac{30080 (- x^{2} + 1500)}{3 x^{4}} \cdot \frac{75}{75} + \frac{376}{225} \cdot \frac{x^{4}}{x^{4}}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.10

Write each expression with a common denominator of $225 x^{4}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.10.4.25.10.1

Multiply $\frac{30080 (- x^{2} + 1500)}{3 x^{4}}$ by $\frac{75}{75}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(\frac{30080 (- x^{2} + 1500) \cdot 75}{3 x^{4} \cdot 75} + \frac{376}{225} \cdot \frac{x^{4}}{x^{4}}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.10.2

Multiply $75$ by $3$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(\frac{30080 (- x^{2} + 1500) \cdot 75}{225 x^{4}} + \frac{376}{225} \cdot \frac{x^{4}}{x^{4}}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.10.3

Multiply $\frac{376}{225}$ by $\frac{x^{4}}{x^{4}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{(\frac{30080 (- x^{2} + 1500) \cdot 75}{225 x^{4}} + \frac{376 x^{4}}{225 x^{4}}) \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.11

Combine the numerators over the common denominator.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{30080 (- x^{2} + 1500) \cdot 75 + 376 x^{4}}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12

Simplify the numerator.

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Step 2.10.4.25.12.1

Factor $376$ out of $30080 (- x^{2} + 1500) \cdot 75 + 376 x^{4}$ .

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Step 2.10.4.25.12.1.1

Factor $376$ out of $30080 (- x^{2} + 1500) \cdot 75$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 (80 (- x^{2} + 1500) \cdot 75) + 376 x^{4}}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.1.2

Factor $376$ out of $376 x^{4}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 (80 (- x^{2} + 1500) \cdot 75) + 376 (x^{4})}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.1.3

Factor $376$ out of $376 (80 (- x^{2} + 1500) \cdot 75) + 376 (x^{4})$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 (80 (- x^{2} + 1500) \cdot 75 + x^{4})}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.2

Apply the distributive property.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 ((80 (- x^{2}) + 80 \cdot 1500) \cdot 75 + x^{4})}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.3

Multiply $- 1$ by $80$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 ((- 80 x^{2} + 80 \cdot 1500) \cdot 75 + x^{4})}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.4

Multiply $80$ by $1500$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 ((- 80 x^{2} + 120000) \cdot 75 + x^{4})}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.5

Apply the distributive property.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 (- 80 x^{2} \cdot 75 + 120000 \cdot 75 + x^{4})}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.6

Multiply $75$ by $- 80$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 (- 6000 x^{2} + 120000 \cdot 75 + x^{4})}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.7

Multiply $120000$ by $75$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 (- 6000 x^{2} + 9000000 + x^{4})}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.8

Reorder terms.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 (x^{4} - 6000 x^{2} + 9000000)}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.9

Rewrite $x^{4} - 6000 x^{2} + 9000000$ in a factored form.

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Step 2.10.4.25.12.9.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 ({(x^{2})}^{2} - 6000 x^{2} + 9000000)}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.9.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 (u^{2} - 6000 u + 9000000)}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.9.3

Factor using the perfect square rule.

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Step 2.10.4.25.12.9.3.1

Rewrite $9000000$ as $3000^{2}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 (u^{2} - 6000 u + 3000^{2})}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.9.3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6000 u = 2 \cdot u \cdot 3000$

Step 2.10.4.25.12.9.3.3

Rewrite the polynomial.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 (u^{2} - 2 \cdot u \cdot 3000 + 3000^{2})}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.9.3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 3000$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 {(u - 3000)}^{2}}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.12.9.4

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

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 {(x^{2} - 3000)}^{2}}{225 x^{4}} \cdot (3375 x^{3})}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.13

Combine exponents.

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Step 2.10.4.25.13.1

Combine $\frac{376 {(x^{2} - 3000)}^{2}}{225 x^{4}}$ and $3375$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{376 {(x^{2} - 3000)}^{2} \cdot 3375}{225 x^{4}} \cdot x^{3}}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.13.2

Multiply $3375$ by $376$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{1269000 {(x^{2} - 3000)}^{2}}{225 x^{4}} \cdot x^{3}}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.13.3

Combine $\frac{1269000 {(x^{2} - 3000)}^{2}}{225 x^{4}}$ and $x^{3}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{1269000 {(x^{2} - 3000)}^{2} x^{3}}{225 x^{4}}}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.14

Reduce the expression $\frac{1269000 {(x^{2} - 3000)}^{2} x^{3}}{225 x^{4}}$ by cancelling the common factors.

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Step 2.10.4.25.14.1

Factor $225$ out of $1269000 {(x^{2} - 3000)}^{2} x^{3}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{225 (5640 {(x^{2} - 3000)}^{2} x^{3})}{225 x^{4}}}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.14.2

Factor $225$ out of $225 x^{4}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{225 (5640 {(x^{2} - 3000)}^{2} x^{3})}{225 (x^{4})}}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.14.3

Cancel the common factor.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{225 (5640 {(x^{2} - 3000)}^{2} x^{3})}{225 x^{4}}}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.14.4

Rewrite the expression.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{5640 {(x^{2} - 3000)}^{2} x^{3}}{x^{4}}}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.15

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 2.10.4.25.15.1

Factor $x^{3}$ out of $5640 {(x^{2} - 3000)}^{2} x^{3}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{x^{3} (5640 {(x^{2} - 3000)}^{2})}{x^{4}}}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.15.2

Cancel the common factors.

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Step 2.10.4.25.15.2.1

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

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{x^{3} (5640 {(x^{2} - 3000)}^{2})}{x^{3} x}}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.15.2.2

Cancel the common factor.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{x^{3} (5640 {(x^{2} - 3000)}^{2})}{x^{3} x}}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.25.15.2.3

Rewrite the expression.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{\frac{5640 {(x^{2} - 3000)}^{2}}{x}}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.26

Multiply the numerator by the reciprocal of the denominator.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{5640 {(x^{2} - 3000)}^{2}}{x} \cdot \frac{1}{{(3000 + x^{2})}^{3}}$

Step 2.10.4.27

Combine.

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{5640 {(x^{2} - 3000)}^{2} \cdot 1}{x {(3000 + x^{2})}^{3}}$

Step 2.10.4.28

Multiply $5640$ by $1$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} + \frac{5640 {(x^{2} - 3000)}^{2}}{x {(3000 + x^{2})}^{3}}$

Step 2.10.5

To write $- \frac{16920000}{x {(3000 + x^{2})}^{2}}$ as a fraction with a common denominator, multiply by $\frac{3000 + x^{2}}{3000 + x^{2}}$ .

$f'' (x) = - \frac{16920000}{x {(3000 + x^{2})}^{2}} \cdot \frac{3000 + x^{2}}{3000 + x^{2}} + \frac{5640 {(x^{2} - 3000)}^{2}}{x {(3000 + x^{2})}^{3}}$

Step 2.10.6

Write each expression with a common denominator of $x {(3000 + x^{2})}^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.10.6.1

Multiply $\frac{16920000}{x {(3000 + x^{2})}^{2}}$ by $\frac{3000 + x^{2}}{3000 + x^{2}}$ .

$f'' (x) = - \frac{16920000 (3000 + x^{2})}{x {(3000 + x^{2})}^{2} (3000 + x^{2})} + \frac{5640 {(x^{2} - 3000)}^{2}}{x {(3000 + x^{2})}^{3}}$

Step 2.10.6.2

Raise $3000 + x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{16920000 (3000 + x^{2})}{x ((3000 + x^{2}) {(3000 + x^{2})}^{2})} + \frac{5640 {(x^{2} - 3000)}^{2}}{x {(3000 + x^{2})}^{3}}$

Step 2.10.6.3

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

$f'' (x) = - \frac{16920000 (3000 + x^{2})}{x {(3000 + x^{2})}^{1 + 2}} + \frac{5640 {(x^{2} - 3000)}^{2}}{x {(3000 + x^{2})}^{3}}$

Step 2.10.6.4

Add $1$ and $2$ .

$f'' (x) = - \frac{16920000 (3000 + x^{2})}{x {(3000 + x^{2})}^{3}} + \frac{5640 {(x^{2} - 3000)}^{2}}{x {(3000 + x^{2})}^{3}}$

Step 2.10.7

Combine the numerators over the common denominator.

$f'' (x) = \frac{- 16920000 (3000 + x^{2}) + 5640 {(x^{2} - 3000)}^{2}}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8

Simplify the numerator.

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Step 2.10.8.1

Factor $5640$ out of $- 16920000 (3000 + x^{2}) + 5640 {(x^{2} - 3000)}^{2}$ .

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Step 2.10.8.1.1

Factor $5640$ out of $- 16920000 (3000 + x^{2})$ .

$f'' (x) = \frac{5640 (- 3000 (3000 + x^{2})) + 5640 {(x^{2} - 3000)}^{2}}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.1.2

Factor $5640$ out of $5640 {(x^{2} - 3000)}^{2}$ .

$f'' (x) = \frac{5640 (- 3000 (3000 + x^{2})) + 5640 ({(x^{2} - 3000)}^{2})}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.1.3

Factor $5640$ out of $5640 (- 3000 (3000 + x^{2})) + 5640 ({(x^{2} - 3000)}^{2})$ .

$f'' (x) = \frac{5640 (- 3000 (3000 + x^{2}) + {(x^{2} - 3000)}^{2})}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.2

Apply the distributive property.

$f'' (x) = \frac{5640 (- 3000 \cdot 3000 - 3000 x^{2} + {(x^{2} - 3000)}^{2})}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.3

Multiply $- 3000$ by $3000$ .

$f'' (x) = \frac{5640 (- 9000000 - 3000 x^{2} + {(x^{2} - 3000)}^{2})}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.4

Rewrite ${(x^{2} - 3000)}^{2}$ as $(x^{2} - 3000) (x^{2} - 3000)$ .

$f'' (x) = \frac{5640 (- 9000000 - 3000 x^{2} + (x^{2} - 3000) (x^{2} - 3000))}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.5

Expand $(x^{2} - 3000) (x^{2} - 3000)$ using the FOIL Method.

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Step 2.10.8.5.1

Apply the distributive property.

$f'' (x) = \frac{5640 (- 9000000 - 3000 x^{2} + x^{2} (x^{2} - 3000) - 3000 (x^{2} - 3000))}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.5.2

Apply the distributive property.

$f'' (x) = \frac{5640 (- 9000000 - 3000 x^{2} + x^{2} x^{2} + x^{2} \cdot - 3000 - 3000 (x^{2} - 3000))}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.5.3

Apply the distributive property.

$f'' (x) = \frac{5640 (- 9000000 - 3000 x^{2} + x^{2} x^{2} + x^{2} \cdot - 3000 - 3000 x^{2} - 3000 \cdot - 3000)}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.6

Simplify and combine like terms.

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Step 2.10.8.6.1

Simplify each term.

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Step 2.10.8.6.1.1

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

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Step 2.10.8.6.1.1.1

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

$f'' (x) = \frac{5640 (- 9000000 - 3000 x^{2} + x^{2 + 2} + x^{2} \cdot - 3000 - 3000 x^{2} - 3000 \cdot - 3000)}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.6.1.1.2

Add $2$ and $2$ .

$f'' (x) = \frac{5640 (- 9000000 - 3000 x^{2} + x^{4} + x^{2} \cdot - 3000 - 3000 x^{2} - 3000 \cdot - 3000)}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.6.1.2

Move $- 3000$ to the left of $x^{2}$ .

$f'' (x) = \frac{5640 (- 9000000 - 3000 x^{2} + x^{4} - 3000 \cdot x^{2} - 3000 x^{2} - 3000 \cdot - 3000)}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.6.1.3

Multiply $- 3000$ by $- 3000$ .

$f'' (x) = \frac{5640 (- 9000000 - 3000 x^{2} + x^{4} - 3000 x^{2} - 3000 x^{2} + 9000000)}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.6.2

Subtract $3000 x^{2}$ from $- 3000 x^{2}$ .

$f'' (x) = \frac{5640 (- 9000000 - 3000 x^{2} + x^{4} - 6000 x^{2} + 9000000)}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.7

Add $- 9000000$ and $9000000$ .

$f'' (x) = \frac{5640 (- 3000 x^{2} + x^{4} - 6000 x^{2} + 0)}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.8

Add $- 3000 x^{2} + x^{4} - 6000 x^{2}$ and $0$ .

$f'' (x) = \frac{5640 (- 3000 x^{2} + x^{4} - 6000 x^{2})}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.9

Subtract $6000 x^{2}$ from $- 3000 x^{2}$ .

$f'' (x) = \frac{5640 (x^{4} - 9000 x^{2})}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.10

Factor $x^{2}$ out of $x^{4} - 9000 x^{2}$ .

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Step 2.10.8.10.1

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

$f'' (x) = \frac{5640 (x^{2} x^{2} - 9000 x^{2})}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.10.2

Factor $x^{2}$ out of $- 9000 x^{2}$ .

$f'' (x) = \frac{5640 (x^{2} x^{2} + x^{2} \cdot - 9000)}{x {(3000 + x^{2})}^{3}}$

Step 2.10.8.10.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 9000$ .

$f'' (x) = \frac{5640 (x^{2} (x^{2} - 9000))}{x {(3000 + x^{2})}^{3}}$

$f'' (x) = \frac{5640 x^{2} (x^{2} - 9000)}{x {(3000 + x^{2})}^{3}}$

Step 2.10.9

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

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Step 2.10.9.1

Factor $x$ out of $5640 x^{2} (x^{2} - 9000)$ .

$f'' (x) = \frac{x (5640 x (x^{2} - 9000))}{x {(3000 + x^{2})}^{3}}$

Step 2.10.9.2

Cancel the common factors.

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Step 2.10.9.2.1

Factor $x$ out of $x {(3000 + x^{2})}^{3}$ .

$f'' (x) = \frac{x (5640 x (x^{2} - 9000))}{x ({(3000 + x^{2})}^{3})}$

Step 2.10.9.2.2

Cancel the common factor.

$f'' (x) = \frac{x (5640 x (x^{2} - 9000))}{x {(3000 + x^{2})}^{3}}$

Step 2.10.9.2.3

Rewrite the expression.

$f'' (x) = \frac{5640 x (x^{2} - 9000)}{{(3000 + x^{2})}^{3}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15}) = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Since $188$ is constant with respect to $x$ , the derivative of $188 {(\frac{200}{x} + \frac{x}{15})}^{- 1}$ with respect to $x$ is $188 \frac{d}{d x} [{(\frac{200}{x} + \frac{x}{15})}^{- 1}]$ .

$188 \frac{d}{d x} [{(\frac{200}{x} + \frac{x}{15})}^{- 1}]$

Step 4.1.2

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 $g (x) = \frac{200}{x} + \frac{x}{15}$ .

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Step 4.1.2.1

To apply the Chain Rule, set $u$ as $\frac{200}{x} + \frac{x}{15}$ .

$188 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [\frac{200}{x} + \frac{x}{15}])$

Step 4.1.2.2

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

$188 (- u^{- 2} \frac{d}{d x} [\frac{200}{x} + \frac{x}{15}])$

Step 4.1.2.3

Replace all occurrences of $u$ with $\frac{200}{x} + \frac{x}{15}$ .

$188 (- {(\frac{200}{x} + \frac{x}{15})}^{- 2} \frac{d}{d x} [\frac{200}{x} + \frac{x}{15}])$

Step 4.1.3

Differentiate.

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Step 4.1.3.1

Multiply $- 1$ by $188$ .

$- 188 ({(\frac{200}{x} + \frac{x}{15})}^{- 2} \frac{d}{d x} [\frac{200}{x} + \frac{x}{15}])$

Step 4.1.3.2

By the Sum Rule, the derivative of $\frac{200}{x} + \frac{x}{15}$ with respect to $x$ is $\frac{d}{d x} [\frac{200}{x}] + \frac{d}{d x} [\frac{x}{15}]$ .

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (\frac{d}{d x} [\frac{200}{x}] + \frac{d}{d x} [\frac{x}{15}])$

Step 4.1.3.3

Since $200$ is constant with respect to $x$ , the derivative of $\frac{200}{x}$ with respect to $x$ is $200 \frac{d}{d x} [\frac{1}{x}]$ .

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (200 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\frac{x}{15}])$

Step 4.1.3.4

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

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (200 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [\frac{x}{15}])$

Step 4.1.3.5

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

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (200 (- x^{- 2}) + \frac{d}{d x} [\frac{x}{15}])$

Step 4.1.3.6

Multiply $- 1$ by $200$ .

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{d}{d x} [\frac{x}{15}])$

Step 4.1.3.7

Since $\frac{1}{15}$ is constant with respect to $x$ , the derivative of $\frac{x}{15}$ with respect to $x$ is $\frac{1}{15} \frac{d}{d x} [x]$ .

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15} \frac{d}{d x} [x])$

Step 4.1.3.8

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

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15} \cdot 1)$

Step 4.1.3.9

Multiply $\frac{1}{15}$ by $1$ .

$f' (x) = - 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15})$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15})$ .

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15})$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15}) = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15}) = 0$

Step 5.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(\frac{200}{x} + \frac{x}{15})}^{- 2} = 0$

$- 200 x^{- 2} + \frac{1}{15} = 0$

Step 5.3

Set ${(\frac{200}{x} + \frac{x}{15})}^{- 2}$ equal to $0$ and solve for $x$ .

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Step 5.3.1

Set ${(\frac{200}{x} + \frac{x}{15})}^{- 2}$ equal to $0$ .

${(\frac{200}{x} + \frac{x}{15})}^{- 2} = 0$

Step 5.3.2

Solve ${(\frac{200}{x} + \frac{x}{15})}^{- 2} = 0$ for $x$ .

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Step 5.3.2.1

Simplify ${(\frac{200}{x} + \frac{x}{15})}^{- 2}$ .

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Step 5.3.2.1.1

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

$\frac{1}{{(\frac{200}{x} + \frac{x}{15})}^{2}} = 0$

Step 5.3.2.1.2

Simplify the denominator.

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Step 5.3.2.1.2.1

To write $\frac{200}{x}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{1}{{(\frac{200}{x} \cdot \frac{15}{15} + \frac{x}{15})}^{2}} = 0$

Step 5.3.2.1.2.2

To write $\frac{x}{15}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{1}{{(\frac{200}{x} \cdot \frac{15}{15} + \frac{x}{15} \cdot \frac{x}{x})}^{2}} = 0$

Step 5.3.2.1.2.3

Write each expression with a common denominator of $x \cdot 15$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.2.1.2.3.1

Multiply $\frac{200}{x}$ by $\frac{15}{15}$ .

$\frac{1}{{(\frac{200 \cdot 15}{x \cdot 15} + \frac{x}{15} \cdot \frac{x}{x})}^{2}} = 0$

Step 5.3.2.1.2.3.2

Multiply $\frac{x}{15}$ by $\frac{x}{x}$ .

$\frac{1}{{(\frac{200 \cdot 15}{x \cdot 15} + \frac{x \cdot x}{15 x})}^{2}} = 0$

Step 5.3.2.1.2.3.3

Reorder the factors of $x \cdot 15$ .

$\frac{1}{{(\frac{200 \cdot 15}{15 x} + \frac{x \cdot x}{15 x})}^{2}} = 0$

Step 5.3.2.1.2.4

Combine the numerators over the common denominator.

$\frac{1}{{(\frac{200 \cdot 15 + x \cdot x}{15 x})}^{2}} = 0$

Step 5.3.2.1.2.5

Simplify the numerator.

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Step 5.3.2.1.2.5.1

Multiply $200$ by $15$ .

$\frac{1}{{(\frac{3000 + x \cdot x}{15 x})}^{2}} = 0$

Step 5.3.2.1.2.5.2

Multiply $x$ by $x$ .

$\frac{1}{{(\frac{3000 + x^{2}}{15 x})}^{2}} = 0$

Step 5.3.2.1.2.6

Apply the product rule to $\frac{3000 + x^{2}}{15 x}$ .

$\frac{1}{\frac{{(3000 + x^{2})}^{2}}{{(15 x)}^{2}}} = 0$

Step 5.3.2.1.2.7

Apply the product rule to $15 x$ .

$\frac{1}{\frac{{(3000 + x^{2})}^{2}}{15^{2} x^{2}}} = 0$

Step 5.3.2.1.2.8

Raise $15$ to the power of $2$ .

$\frac{1}{\frac{{(3000 + x^{2})}^{2}}{225 x^{2}}} = 0$

Step 5.3.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{225 x^{2}}{{(3000 + x^{2})}^{2}} = 0$

Step 5.3.2.1.4

Multiply $\frac{225 x^{2}}{{(3000 + x^{2})}^{2}}$ by $1$ .

$\frac{225 x^{2}}{{(3000 + x^{2})}^{2}} = 0$

Step 5.3.2.2

Set the numerator equal to zero.

$225 x^{2} = 0$

Step 5.3.2.3

Solve the equation for $x$ .

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Step 5.3.2.3.1

Divide each term in $225 x^{2} = 0$ by $225$ and simplify.

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Step 5.3.2.3.1.1

Divide each term in $225 x^{2} = 0$ by $225$ .

$\frac{225 x^{2}}{225} = \frac{0}{225}$

Step 5.3.2.3.1.2

Simplify the left side.

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Step 5.3.2.3.1.2.1

Cancel the common factor of $225$ .

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Step 5.3.2.3.1.2.1.1

Cancel the common factor.

$\frac{225 x^{2}}{225} = \frac{0}{225}$

Step 5.3.2.3.1.2.1.2

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

$x^{2} = \frac{0}{225}$

Step 5.3.2.3.1.3

Simplify the right side.

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Step 5.3.2.3.1.3.1

Divide $0$ by $225$ .

$x^{2} = 0$

Step 5.3.2.3.2

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

$x = \pm \sqrt{0}$

Step 5.3.2.3.3

Simplify $\pm \sqrt{0}$ .

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Step 5.3.2.3.3.1

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

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

Step 5.3.2.3.3.2

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

$x = \pm 0$

Step 5.3.2.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5.4

Set $- 200 x^{- 2} + \frac{1}{15}$ equal to $0$ and solve for $x$ .

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Step 5.4.1

Set $- 200 x^{- 2} + \frac{1}{15}$ equal to $0$ .

$- 200 x^{- 2} + \frac{1}{15} = 0$

Step 5.4.2

Solve $- 200 x^{- 2} + \frac{1}{15} = 0$ for $x$ .

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

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

$- 200 \frac{1}{x^{2}} + \frac{1}{15} = 0$

Step 5.4.2.1.2

Combine $- 200$ and $\frac{1}{x^{2}}$ .

$\frac{- 200}{x^{2}} + \frac{1}{15} = 0$

Step 5.4.2.1.3

Move the negative in front of the fraction.

$- \frac{200}{x^{2}} + \frac{1}{15} = 0$

Step 5.4.2.2

Subtract $\frac{1}{15}$ from both sides of the equation.

$- \frac{200}{x^{2}} = - \frac{1}{15}$

Step 5.4.2.3

Find the LCD of the terms in the equation.

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Step 5.4.2.3.1

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

$x^{2}, 15$

Step 5.4.2.3.2

Since $x^{2}, 15$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 15$ then find LCM for the variable part $x^{2}$ .

Step 5.4.2.3.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 5.4.2.3.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 5.4.2.3.5

$15$ has factors of $3$ and $5$ .

$3 \cdot 5$

Step 5.4.2.3.6

Multiply $3$ by $5$ .

$15$

Step 5.4.2.3.7

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

Step 5.4.2.3.8

The LCM of $x^{2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x$

Step 5.4.2.3.9

Multiply $x$ by $x$ .

$x^{2}$

Step 5.4.2.3.10

The LCM for $x^{2}, 15$ is the numeric part $15$ multiplied by the variable part.

$15 x^{2}$

Step 5.4.2.4

Multiply each term in $- \frac{200}{x^{2}} = - \frac{1}{15}$ by $15 x^{2}$ to eliminate the fractions.

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Step 5.4.2.4.1

Multiply each term in $- \frac{200}{x^{2}} = - \frac{1}{15}$ by $15 x^{2}$ .

$- \frac{200}{x^{2}} (15 x^{2}) = - \frac{1}{15} (15 x^{2})$

Step 5.4.2.4.2

Simplify the left side.

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Step 5.4.2.4.2.1

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

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Step 5.4.2.4.2.1.1

Move the leading negative in $- \frac{200}{x^{2}}$ into the numerator.

$\frac{- 200}{x^{2}} (15 x^{2}) = - \frac{1}{15} (15 x^{2})$

Step 5.4.2.4.2.1.2

Factor $x^{2}$ out of $15 x^{2}$ .

$\frac{- 200}{x^{2}} (x^{2} \cdot 15) = - \frac{1}{15} (15 x^{2})$

Step 5.4.2.4.2.1.3

Cancel the common factor.

$\frac{- 200}{x^{2}} (x^{2} \cdot 15) = - \frac{1}{15} (15 x^{2})$

Step 5.4.2.4.2.1.4

Rewrite the expression.

$- 200 \cdot 15 = - \frac{1}{15} (15 x^{2})$

Step 5.4.2.4.2.2

Multiply $- 200$ by $15$ .

$- 3000 = - \frac{1}{15} (15 x^{2})$

Step 5.4.2.4.3

Simplify the right side.

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Step 5.4.2.4.3.1

Cancel the common factor of $15$ .

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Step 5.4.2.4.3.1.1

Move the leading negative in $- \frac{1}{15}$ into the numerator.

$- 3000 = \frac{- 1}{15} (15 x^{2})$

Step 5.4.2.4.3.1.2

Factor $15$ out of $15 x^{2}$ .

$- 3000 = \frac{- 1}{15} (15 (x^{2}))$

Step 5.4.2.4.3.1.3

Cancel the common factor.

$- 3000 = \frac{- 1}{15} (15 x^{2})$

Step 5.4.2.4.3.1.4

Rewrite the expression.

$- 3000 = - x^{2}$

Step 5.4.2.5

Solve the equation.

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Step 5.4.2.5.1

Rewrite the equation as $- x^{2} = - 3000$ .

$- x^{2} = - 3000$

Step 5.4.2.5.2

Divide each term in $- x^{2} = - 3000$ by $- 1$ and simplify.

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Step 5.4.2.5.2.1

Divide each term in $- x^{2} = - 3000$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 3000}{- 1}$

Step 5.4.2.5.2.2

Simplify the left side.

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Step 5.4.2.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 3000}{- 1}$

Step 5.4.2.5.2.2.2

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

$x^{2} = \frac{- 3000}{- 1}$

Step 5.4.2.5.2.3

Simplify the right side.

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Step 5.4.2.5.2.3.1

Divide $- 3000$ by $- 1$ .

$x^{2} = 3000$

Step 5.4.2.5.3

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

$x = \pm \sqrt{3000}$

Step 5.4.2.5.4

Simplify $\pm \sqrt{3000}$ .

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Step 5.4.2.5.4.1

Rewrite $3000$ as $10^{2} \cdot 30$ .

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Step 5.4.2.5.4.1.1

Factor $100$ out of $3000$ .

$x = \pm \sqrt{100 (30)}$

Step 5.4.2.5.4.1.2

Rewrite $100$ as $10^{2}$ .

$x = \pm \sqrt{10^{2} \cdot 30}$

Step 5.4.2.5.4.2

Pull terms out from under the radical.

$x = \pm 10 \sqrt{30}$

Step 5.4.2.5.5

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

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Step 5.4.2.5.5.1

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

$x = 10 \sqrt{30}$

Step 5.4.2.5.5.2

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

$x = - 10 \sqrt{30}$

Step 5.4.2.5.5.3

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

$x = 10 \sqrt{30}, - 10 \sqrt{30}$

Step 5.5

The final solution is all the values that make $- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15}) = 0$ true.

$x = 0, 10 \sqrt{30}, - 10 \sqrt{30}$

Step 5.6

Exclude the solutions that do not make $- 188 {(\frac{200}{x} + \frac{x}{15})}^{- 2} (- 200 x^{- 2} + \frac{1}{15}) = 0$ true.

$x = 10 \sqrt{30}, - 10 \sqrt{30}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{200}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 6.2

Set the base in ${(\frac{200}{x} + \frac{x}{15})}^{- 2}$ equal to $0$ to find where the expression is undefined.

$\frac{200}{x} + \frac{x}{15} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

Find the LCD of the terms in the equation.

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Step 6.3.1.1

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

$x, 15, 1$

Step 6.3.1.2

Since $x, 15, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 15, 1$ then find LCM for the variable part $x^{1}$ .

Step 6.3.1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 6.3.1.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 6.3.1.5

$15$ has factors of $3$ and $5$ .

$3 \cdot 5$

Step 6.3.1.6

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 6.3.1.7

The LCM of $1, 15, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3 \cdot 5$

Step 6.3.1.8

Multiply $3$ by $5$ .

$15$

Step 6.3.1.9

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 6.3.1.10

The LCM of $x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x$

Step 6.3.1.11

The LCM for $x, 15, 1$ is the numeric part $15$ multiplied by the variable part.

$15 x$

Step 6.3.2

Multiply each term in $\frac{200}{x} + \frac{x}{15} = 0$ by $15 x$ to eliminate the fractions.

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Step 6.3.2.1

Multiply each term in $\frac{200}{x} + \frac{x}{15} = 0$ by $15 x$ .

$\frac{200}{x} (15 x) + \frac{x}{15} (15 x) = 0 (15 x)$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify each term.

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Step 6.3.2.2.1.1

Rewrite using the commutative property of multiplication.

$15 \frac{200}{x} x + \frac{x}{15} (15 x) = 0 (15 x)$

Step 6.3.2.2.1.2

Multiply $15 \frac{200}{x}$ .

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Step 6.3.2.2.1.2.1

Combine $15$ and $\frac{200}{x}$ .

$\frac{15 \cdot 200}{x} x + \frac{x}{15} (15 x) = 0 (15 x)$

Step 6.3.2.2.1.2.2

Multiply $15$ by $200$ .

$\frac{3000}{x} x + \frac{x}{15} (15 x) = 0 (15 x)$

Step 6.3.2.2.1.3

Cancel the common factor of $x$ .

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Step 6.3.2.2.1.3.1

Cancel the common factor.

$\frac{3000}{x} x + \frac{x}{15} (15 x) = 0 (15 x)$

Step 6.3.2.2.1.3.2

Rewrite the expression.

$3000 + \frac{x}{15} (15 x) = 0 (15 x)$

Step 6.3.2.2.1.4

Rewrite using the commutative property of multiplication.

$3000 + 15 \frac{x}{15} x = 0 (15 x)$

Step 6.3.2.2.1.5

Cancel the common factor of $15$ .

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Step 6.3.2.2.1.5.1

Cancel the common factor.

$3000 + 15 \frac{x}{15} x = 0 (15 x)$

Step 6.3.2.2.1.5.2

Rewrite the expression.

$3000 + x \cdot x = 0 (15 x)$

Step 6.3.2.2.1.6

Multiply $x$ by $x$ .

$3000 + x^{2} = 0 (15 x)$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Multiply $0 (15 x)$ .

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Step 6.3.2.3.1.1

Multiply $15$ by $0$ .

$3000 + x^{2} = 0 x$

Step 6.3.2.3.1.2

Multiply $0$ by $x$ .

$3000 + x^{2} = 0$

Step 6.3.3

Solve the equation.

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Step 6.3.3.1

Subtract $3000$ from both sides of the equation.

$x^{2} = - 3000$

Step 6.3.3.2

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

$x = \pm \sqrt{- 3000}$

Step 6.3.3.3

Simplify $\pm \sqrt{- 3000}$ .

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Step 6.3.3.3.1

Rewrite $- 3000$ as $- 1 (3000)$ .

$x = \pm \sqrt{- 1 (3000)}$

Step 6.3.3.3.2

Rewrite $\sqrt{- 1 (3000)}$ as $\sqrt{- 1} \cdot \sqrt{3000}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{3000}$

Step 6.3.3.3.3

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

$x = \pm i \cdot \sqrt{3000}$

Step 6.3.3.3.4

Rewrite $3000$ as $10^{2} \cdot 30$ .

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Step 6.3.3.3.4.1

Factor $100$ out of $3000$ .

$x = \pm i \cdot \sqrt{100 (30)}$

Step 6.3.3.3.4.2

Rewrite $100$ as $10^{2}$ .

$x = \pm i \cdot \sqrt{10^{2} \cdot 30}$

Step 6.3.3.3.5

Pull terms out from under the radical.

$x = \pm i \cdot (10 \sqrt{30})$

Step 6.3.3.3.6

Move $10$ to the left of $i$ .

$x = \pm 10 i \sqrt{30}$

Step 6.3.3.4

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

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Step 6.3.3.4.1

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

$x = 10 i \sqrt{30}$

Step 6.3.3.4.2

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

$x = - 10 i \sqrt{30}$

Step 6.3.3.4.3

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

$x = 10 i \sqrt{30}, - 10 i \sqrt{30}$

Step 6.4

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 $0$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = 10 \sqrt{30}, - 10 \sqrt{30}$

Step 8

Evaluate the second derivative at $x = 10 \sqrt{30}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{5640 (10 \sqrt{30}) ({(10 \sqrt{30})}^{2} - 9000)}{{(3000 + {(10 \sqrt{30})}^{2})}^{3}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Multiply $5640$ by $10$ .

$\frac{56400 \sqrt{30} ({(10 \sqrt{30})}^{2} - 9000)}{{(3000 + {(10 \sqrt{30})}^{2})}^{3}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Apply the product rule to $10 \sqrt{30}$ .

$\frac{56400 \sqrt{30} ({(10 \sqrt{30})}^{2} - 9000)}{{(3000 + 10^{2} {\sqrt{30}}^{2})}^{3}}$

Step 9.2.2

Raise $10$ to the power of $2$ .

$\frac{56400 \sqrt{30} ({(10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 {\sqrt{30}}^{2})}^{3}}$

Step 9.2.3

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

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Step 9.2.3.1

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

$\frac{56400 \sqrt{30} ({(10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 {(30^{\frac{1}{2}})}^{2})}^{3}}$

Step 9.2.3.2

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

$\frac{56400 \sqrt{30} ({(10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 \cdot 30^{\frac{1}{2} \cdot 2})}^{3}}$

Step 9.2.3.3

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

$\frac{56400 \sqrt{30} ({(10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 \cdot 30^{\frac{2}{2}})}^{3}}$

Step 9.2.3.4

Cancel the common factor of $2$ .

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Step 9.2.3.4.1

Cancel the common factor.

$\frac{56400 \sqrt{30} ({(10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 \cdot 30^{\frac{2}{2}})}^{3}}$

Step 9.2.3.4.2

Rewrite the expression.

$\frac{56400 \sqrt{30} ({(10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 \cdot 30^{1})}^{3}}$

Step 9.2.3.5

Evaluate the exponent.

$\frac{56400 \sqrt{30} ({(10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 \cdot 30)}^{3}}$

Step 9.2.4

Multiply $100$ by $30$ .

$\frac{56400 \sqrt{30} ({(10 \sqrt{30})}^{2} - 9000)}{{(3000 + 3000)}^{3}}$

Step 9.2.5

Add $3000$ and $3000$ .

$\frac{56400 \sqrt{30} ({(10 \sqrt{30})}^{2} - 9000)}{6000^{3}}$

Step 9.2.6

Raise $6000$ to the power of $3$ .

$\frac{56400 \sqrt{30} ({(10 \sqrt{30})}^{2} - 9000)}{216000000000}$

Step 9.3

Simplify the numerator.

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Step 9.3.1

Apply the product rule to $10 \sqrt{30}$ .

$\frac{56400 \sqrt{30} (10^{2} {\sqrt{30}}^{2} - 9000)}{216000000000}$

Step 9.3.2

Raise $10$ to the power of $2$ .

$\frac{56400 \sqrt{30} (100 {\sqrt{30}}^{2} - 9000)}{216000000000}$

Step 9.3.3

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

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Step 9.3.3.1

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

$\frac{56400 \sqrt{30} (100 {(30^{\frac{1}{2}})}^{2} - 9000)}{216000000000}$

Step 9.3.3.2

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

$\frac{56400 \sqrt{30} (100 \cdot 30^{\frac{1}{2} \cdot 2} - 9000)}{216000000000}$

Step 9.3.3.3

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

$\frac{56400 \sqrt{30} (100 \cdot 30^{\frac{2}{2}} - 9000)}{216000000000}$

Step 9.3.3.4

Cancel the common factor of $2$ .

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Step 9.3.3.4.1

Cancel the common factor.

$\frac{56400 \sqrt{30} (100 \cdot 30^{\frac{2}{2}} - 9000)}{216000000000}$

Step 9.3.3.4.2

Rewrite the expression.

$\frac{56400 \sqrt{30} (100 \cdot 30^{1} - 9000)}{216000000000}$

Step 9.3.3.5

Evaluate the exponent.

$\frac{56400 \sqrt{30} (100 \cdot 30 - 9000)}{216000000000}$

Step 9.3.4

Multiply $100$ by $30$ .

$\frac{56400 \sqrt{30} (3000 - 9000)}{216000000000}$

Step 9.3.5

Subtract $9000$ from $3000$ .

$\frac{56400 \sqrt{30} \cdot - 6000}{216000000000}$

Step 9.3.6

Multiply $- 6000$ by $56400$ .

$\frac{- 338400000 \sqrt{30}}{216000000000}$

Step 9.4

Reduce the expression by cancelling the common factors.

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Step 9.4.1

Cancel the common factor of $- 338400000$ and $216000000000$ .

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Step 9.4.1.1

Factor $7200000$ out of $- 338400000 \sqrt{30}$ .

$\frac{7200000 (- 47 \sqrt{30})}{216000000000}$

Step 9.4.1.2

Cancel the common factors.

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Step 9.4.1.2.1

Factor $7200000$ out of $216000000000$ .

$\frac{7200000 (- 47 \sqrt{30})}{7200000 (30000)}$

Step 9.4.1.2.2

Cancel the common factor.

$\frac{7200000 (- 47 \sqrt{30})}{7200000 \cdot 30000}$

Step 9.4.1.2.3

Rewrite the expression.

$\frac{- 47 \sqrt{30}}{30000}$

Step 9.4.2

Move the negative in front of the fraction.

$- \frac{47 \sqrt{30}}{30000}$

Step 10

$x = 10 \sqrt{30}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 10 \sqrt{30}$ is a local maximum

Step 11

Find the y-value when $x = 10 \sqrt{30}$ .

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Step 11.1

Replace the variable $x$ with $10 \sqrt{30}$ in the expression.

$f (10 \sqrt{30}) = 188 {(\frac{200}{10 \sqrt{30}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Cancel the common factor of $200$ and $10$ .

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Step 11.2.1.1.1

Factor $10$ out of $200$ .

$f (10 \sqrt{30}) = 188 {(\frac{10 \cdot 20}{10 \sqrt{30}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.1.2

Cancel the common factors.

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Step 11.2.1.1.2.1

Factor $10$ out of $10 \sqrt{30}$ .

$f (10 \sqrt{30}) = 188 {(\frac{10 \cdot 20}{10 (\sqrt{30})} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.1.2.2

Cancel the common factor.

$f (10 \sqrt{30}) = 188 {(\frac{10 \cdot 20}{10 \sqrt{30}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.1.2.3

Rewrite the expression.

$f (10 \sqrt{30}) = 188 {(\frac{20}{\sqrt{30}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.2

Multiply $\frac{20}{\sqrt{30}}$ by $\frac{\sqrt{30}}{\sqrt{30}}$ .

$f (10 \sqrt{30}) = 188 {(\frac{20}{\sqrt{30}} \cdot \frac{\sqrt{30}}{\sqrt{30}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.3

Combine and simplify the denominator.

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Step 11.2.1.3.1

Multiply $\frac{20}{\sqrt{30}}$ by $\frac{\sqrt{30}}{\sqrt{30}}$ .

$f (10 \sqrt{30}) = 188 {(\frac{20 \sqrt{30}}{\sqrt{30} \sqrt{30}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.3.2

Raise $\sqrt{30}$ to the power of $1$ .

$f (10 \sqrt{30}) = 188 {(\frac{20 \sqrt{30}}{\sqrt{30} \sqrt{30}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.3.3

Raise $\sqrt{30}$ to the power of $1$ .

$f (10 \sqrt{30}) = 188 {(\frac{20 \sqrt{30}}{\sqrt{30} \sqrt{30}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.3.4

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

$f (10 \sqrt{30}) = 188 {(\frac{20 \sqrt{30}}{{\sqrt{30}}^{1 + 1}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.3.5

Add $1$ and $1$ .

$f (10 \sqrt{30}) = 188 {(\frac{20 \sqrt{30}}{{\sqrt{30}}^{2}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.3.6

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

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Step 11.2.1.3.6.1

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

$f (10 \sqrt{30}) = 188 {(\frac{20 \sqrt{30}}{{(30^{\frac{1}{2}})}^{2}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.3.6.2

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

$f (10 \sqrt{30}) = 188 {(\frac{20 \sqrt{30}}{30^{\frac{1}{2} \cdot 2}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.3.6.3

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

$f (10 \sqrt{30}) = 188 {(\frac{20 \sqrt{30}}{30^{\frac{2}{2}}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.3.6.4

Cancel the common factor of $2$ .

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Step 11.2.1.3.6.4.1

Cancel the common factor.

$f (10 \sqrt{30}) = 188 {(\frac{20 \sqrt{30}}{30^{\frac{2}{2}}} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.3.6.4.2

Rewrite the expression.

$f (10 \sqrt{30}) = 188 {(\frac{20 \sqrt{30}}{30} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.3.6.5

Evaluate the exponent.

$f (10 \sqrt{30}) = 188 {(\frac{20 \sqrt{30}}{30} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.4

Cancel the common factor of $20$ and $30$ .

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Step 11.2.1.4.1

Factor $10$ out of $20 \sqrt{30}$ .

$f (10 \sqrt{30}) = 188 {(\frac{10 (2 \sqrt{30})}{30} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.4.2

Cancel the common factors.

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Step 11.2.1.4.2.1

Factor $10$ out of $30$ .

$f (10 \sqrt{30}) = 188 {(\frac{10 (2 \sqrt{30})}{10 (3)} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.4.2.2

Cancel the common factor.

$f (10 \sqrt{30}) = 188 {(\frac{10 (2 \sqrt{30})}{10 \cdot 3} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.4.2.3

Rewrite the expression.

$f (10 \sqrt{30}) = 188 {(\frac{2 \sqrt{30}}{3} + \frac{10 \sqrt{30}}{15})}^{- 1}$

Step 11.2.1.5

Cancel the common factor of $10$ and $15$ .

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Step 11.2.1.5.1

Factor $5$ out of $10 \sqrt{30}$ .

$f (10 \sqrt{30}) = 188 {(\frac{2 \sqrt{30}}{3} + \frac{5 (2 \sqrt{30})}{15})}^{- 1}$

Step 11.2.1.5.2

Cancel the common factors.

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Step 11.2.1.5.2.1

Factor $5$ out of $15$ .

$f (10 \sqrt{30}) = 188 {(\frac{2 \sqrt{30}}{3} + \frac{5 (2 \sqrt{30})}{5 (3)})}^{- 1}$

Step 11.2.1.5.2.2

Cancel the common factor.

$f (10 \sqrt{30}) = 188 {(\frac{2 \sqrt{30}}{3} + \frac{5 (2 \sqrt{30})}{5 \cdot 3})}^{- 1}$

Step 11.2.1.5.2.3

Rewrite the expression.

$f (10 \sqrt{30}) = 188 {(\frac{2 \sqrt{30}}{3} + \frac{2 \sqrt{30}}{3})}^{- 1}$

Step 11.2.2

Simplify terms.

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Step 11.2.2.1

Combine the numerators over the common denominator.

$f (10 \sqrt{30}) = 188 {(\frac{2 \sqrt{30} + 2 \sqrt{30}}{3})}^{- 1}$

Step 11.2.2.2

Add $2 \sqrt{30}$ and $2 \sqrt{30}$ .

$f (10 \sqrt{30}) = 188 {(\frac{4 \sqrt{30}}{3})}^{- 1}$

Step 11.2.3

Change the sign of the exponent by rewriting the base as its reciprocal.

$f (10 \sqrt{30}) = 188 (\frac{3}{4 \sqrt{30}})$

Step 11.2.4

Cancel the common factor of $4$ .

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Step 11.2.4.1

Factor $4$ out of $188$ .

$f (10 \sqrt{30}) = 4 (47) (\frac{3}{4 \sqrt{30}})$

Step 11.2.4.2

Factor $4$ out of $4 \sqrt{30}$ .

$f (10 \sqrt{30}) = 4 (47) (\frac{3}{4 (\sqrt{30})})$

Step 11.2.4.3

Cancel the common factor.

$f (10 \sqrt{30}) = 4 \cdot (47 (\frac{3}{4 \sqrt{30}}))$

Step 11.2.4.4

Rewrite the expression.

$f (10 \sqrt{30}) = 47 (\frac{3}{\sqrt{30}})$

Step 11.2.5

Combine $47$ and $\frac{3}{\sqrt{30}}$ .

$f (10 \sqrt{30}) = \frac{47 \cdot 3}{\sqrt{30}}$

Step 11.2.6

Multiply $47$ by $3$ .

$f (10 \sqrt{30}) = \frac{141}{\sqrt{30}}$

Step 11.2.7

Multiply $\frac{141}{\sqrt{30}}$ by $\frac{\sqrt{30}}{\sqrt{30}}$ .

$f (10 \sqrt{30}) = \frac{141}{\sqrt{30}} \cdot \frac{\sqrt{30}}{\sqrt{30}}$

Step 11.2.8

Combine and simplify the denominator.

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Step 11.2.8.1

Multiply $\frac{141}{\sqrt{30}}$ by $\frac{\sqrt{30}}{\sqrt{30}}$ .

$f (10 \sqrt{30}) = \frac{141 \sqrt{30}}{\sqrt{30} \sqrt{30}}$

Step 11.2.8.2

Raise $\sqrt{30}$ to the power of $1$ .

$f (10 \sqrt{30}) = \frac{141 \sqrt{30}}{\sqrt{30} \sqrt{30}}$

Step 11.2.8.3

Raise $\sqrt{30}$ to the power of $1$ .

$f (10 \sqrt{30}) = \frac{141 \sqrt{30}}{\sqrt{30} \sqrt{30}}$

Step 11.2.8.4

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

$f (10 \sqrt{30}) = \frac{141 \sqrt{30}}{{\sqrt{30}}^{1 + 1}}$

Step 11.2.8.5

Add $1$ and $1$ .

$f (10 \sqrt{30}) = \frac{141 \sqrt{30}}{{\sqrt{30}}^{2}}$

Step 11.2.8.6

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

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Step 11.2.8.6.1

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

$f (10 \sqrt{30}) = \frac{141 \sqrt{30}}{{(30^{\frac{1}{2}})}^{2}}$

Step 11.2.8.6.2

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

$f (10 \sqrt{30}) = \frac{141 \sqrt{30}}{30^{\frac{1}{2} \cdot 2}}$

Step 11.2.8.6.3

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

$f (10 \sqrt{30}) = \frac{141 \sqrt{30}}{30^{\frac{2}{2}}}$

Step 11.2.8.6.4

Cancel the common factor of $2$ .

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Step 11.2.8.6.4.1

Cancel the common factor.

$f (10 \sqrt{30}) = \frac{141 \sqrt{30}}{30^{\frac{2}{2}}}$

Step 11.2.8.6.4.2

Rewrite the expression.

$f (10 \sqrt{30}) = \frac{141 \sqrt{30}}{30}$

Step 11.2.8.6.5

Evaluate the exponent.

$f (10 \sqrt{30}) = \frac{141 \sqrt{30}}{30}$

Step 11.2.9

Cancel the common factor of $141$ and $30$ .

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Step 11.2.9.1

Factor $3$ out of $141 \sqrt{30}$ .

$f (10 \sqrt{30}) = \frac{3 (47 \sqrt{30})}{30}$

Step 11.2.9.2

Cancel the common factors.

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Step 11.2.9.2.1

Factor $3$ out of $30$ .

$f (10 \sqrt{30}) = \frac{3 (47 \sqrt{30})}{3 (10)}$

Step 11.2.9.2.2

Cancel the common factor.

$f (10 \sqrt{30}) = \frac{3 (47 \sqrt{30})}{3 \cdot 10}$

Step 11.2.9.2.3

Rewrite the expression.

$f (10 \sqrt{30}) = \frac{47 \sqrt{30}}{10}$

Step 11.2.10

The final answer is $\frac{47 \sqrt{30}}{10}$ .

$y = \frac{47 \sqrt{30}}{10}$

Step 12

Evaluate the second derivative at $x = - 10 \sqrt{30}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{5640 (- 10 \sqrt{30}) ({(- 10 \sqrt{30})}^{2} - 9000)}{{(3000 + {(- 10 \sqrt{30})}^{2})}^{3}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Multiply $- 10$ by $5640$ .

$\frac{- 56400 \sqrt{30} ({(- 10 \sqrt{30})}^{2} - 9000)}{{(3000 + {(- 10 \sqrt{30})}^{2})}^{3}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Apply the product rule to $- 10 \sqrt{30}$ .

$\frac{- 56400 \sqrt{30} ({(- 10 \sqrt{30})}^{2} - 9000)}{{(3000 + {(- 10)}^{2} {\sqrt{30}}^{2})}^{3}}$

Step 13.2.2

Raise $- 10$ to the power of $2$ .

$\frac{- 56400 \sqrt{30} ({(- 10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 {\sqrt{30}}^{2})}^{3}}$

Step 13.2.3

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

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Step 13.2.3.1

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

$\frac{- 56400 \sqrt{30} ({(- 10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 {(30^{\frac{1}{2}})}^{2})}^{3}}$

Step 13.2.3.2

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

$\frac{- 56400 \sqrt{30} ({(- 10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 \cdot 30^{\frac{1}{2} \cdot 2})}^{3}}$

Step 13.2.3.3

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

$\frac{- 56400 \sqrt{30} ({(- 10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 \cdot 30^{\frac{2}{2}})}^{3}}$

Step 13.2.3.4

Cancel the common factor of $2$ .

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Step 13.2.3.4.1

Cancel the common factor.

$\frac{- 56400 \sqrt{30} ({(- 10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 \cdot 30^{\frac{2}{2}})}^{3}}$

Step 13.2.3.4.2

Rewrite the expression.

$\frac{- 56400 \sqrt{30} ({(- 10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 \cdot 30^{1})}^{3}}$

Step 13.2.3.5

Evaluate the exponent.

$\frac{- 56400 \sqrt{30} ({(- 10 \sqrt{30})}^{2} - 9000)}{{(3000 + 100 \cdot 30)}^{3}}$

Step 13.2.4

Multiply $100$ by $30$ .

$\frac{- 56400 \sqrt{30} ({(- 10 \sqrt{30})}^{2} - 9000)}{{(3000 + 3000)}^{3}}$

Step 13.2.5

Add $3000$ and $3000$ .

$\frac{- 56400 \sqrt{30} ({(- 10 \sqrt{30})}^{2} - 9000)}{6000^{3}}$

Step 13.2.6

Raise $6000$ to the power of $3$ .

$\frac{- 56400 \sqrt{30} ({(- 10 \sqrt{30})}^{2} - 9000)}{216000000000}$

Step 13.3

Simplify the numerator.

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Step 13.3.1

Apply the product rule to $- 10 \sqrt{30}$ .

$\frac{- 56400 \sqrt{30} ({(- 10)}^{2} {\sqrt{30}}^{2} - 9000)}{216000000000}$

Step 13.3.2

Raise $- 10$ to the power of $2$ .

$\frac{- 56400 \sqrt{30} (100 {\sqrt{30}}^{2} - 9000)}{216000000000}$

Step 13.3.3

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

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Step 13.3.3.1

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

$\frac{- 56400 \sqrt{30} (100 {(30^{\frac{1}{2}})}^{2} - 9000)}{216000000000}$

Step 13.3.3.2

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

$\frac{- 56400 \sqrt{30} (100 \cdot 30^{\frac{1}{2} \cdot 2} - 9000)}{216000000000}$

Step 13.3.3.3

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

$\frac{- 56400 \sqrt{30} (100 \cdot 30^{\frac{2}{2}} - 9000)}{216000000000}$

Step 13.3.3.4

Cancel the common factor of $2$ .

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Step 13.3.3.4.1

Cancel the common factor.

$\frac{- 56400 \sqrt{30} (100 \cdot 30^{\frac{2}{2}} - 9000)}{216000000000}$

Step 13.3.3.4.2

Rewrite the expression.

$\frac{- 56400 \sqrt{30} (100 \cdot 30^{1} - 9000)}{216000000000}$

Step 13.3.3.5

Evaluate the exponent.

$\frac{- 56400 \sqrt{30} (100 \cdot 30 - 9000)}{216000000000}$

Step 13.3.4

Multiply $100$ by $30$ .

$\frac{- 56400 \sqrt{30} (3000 - 9000)}{216000000000}$

Step 13.3.5

Subtract $9000$ from $3000$ .

$\frac{- 56400 \sqrt{30} \cdot - 6000}{216000000000}$

Step 13.3.6

Multiply $- 6000$ by $- 56400$ .

$\frac{338400000 \sqrt{30}}{216000000000}$

Step 13.4

Cancel the common factor of $338400000$ and $216000000000$ .

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Step 13.4.1

Factor $7200000$ out of $338400000 \sqrt{30}$ .

$\frac{7200000 (47 \sqrt{30})}{216000000000}$

Step 13.4.2

Cancel the common factors.

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Step 13.4.2.1

Factor $7200000$ out of $216000000000$ .

$\frac{7200000 (47 \sqrt{30})}{7200000 (30000)}$

Step 13.4.2.2

Cancel the common factor.

$\frac{7200000 (47 \sqrt{30})}{7200000 \cdot 30000}$

Step 13.4.2.3

Rewrite the expression.

$\frac{47 \sqrt{30}}{30000}$

Step 14

$x = - 10 \sqrt{30}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 10 \sqrt{30}$ is a local minimum

Step 15

Find the y-value when $x = - 10 \sqrt{30}$ .

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Step 15.1

Replace the variable $x$ with $- 10 \sqrt{30}$ in the expression.

$f (- 10 \sqrt{30}) = 188 {(\frac{200}{- 10 \sqrt{30}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Cancel the common factor of $200$ and $- 10$ .

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Step 15.2.1.1.1

Factor $10$ out of $200$ .

$f (- 10 \sqrt{30}) = 188 {(\frac{10 (20)}{- 10 \sqrt{30}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.1.2

Cancel the common factors.

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Step 15.2.1.1.2.1

Factor $10$ out of $- 10 \sqrt{30}$ .

$f (- 10 \sqrt{30}) = 188 {(\frac{10 (20)}{10 (- \sqrt{30})} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.1.2.2

Cancel the common factor.

$f (- 10 \sqrt{30}) = 188 {(\frac{10 \cdot 20}{10 (- \sqrt{30})} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.1.2.3

Rewrite the expression.

$f (- 10 \sqrt{30}) = 188 {(\frac{20}{- \sqrt{30}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.2

Move the negative in front of the fraction.

$f (- 10 \sqrt{30}) = 188 {(- \frac{20}{\sqrt{30}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.3

Multiply $\frac{20}{\sqrt{30}}$ by $\frac{\sqrt{30}}{\sqrt{30}}$ .

$f (- 10 \sqrt{30}) = 188 {(- (\frac{20}{\sqrt{30}} \cdot \frac{\sqrt{30}}{\sqrt{30}}) + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.4

Combine and simplify the denominator.

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Step 15.2.1.4.1

Multiply $\frac{20}{\sqrt{30}}$ by $\frac{\sqrt{30}}{\sqrt{30}}$ .

$f (- 10 \sqrt{30}) = 188 {(- \frac{20 \sqrt{30}}{\sqrt{30} \sqrt{30}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.4.2

Raise $\sqrt{30}$ to the power of $1$ .

$f (- 10 \sqrt{30}) = 188 {(- \frac{20 \sqrt{30}}{\sqrt{30} \sqrt{30}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.4.3

Raise $\sqrt{30}$ to the power of $1$ .

$f (- 10 \sqrt{30}) = 188 {(- \frac{20 \sqrt{30}}{\sqrt{30} \sqrt{30}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.4.4

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

$f (- 10 \sqrt{30}) = 188 {(- \frac{20 \sqrt{30}}{{\sqrt{30}}^{1 + 1}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.4.5

Add $1$ and $1$ .

$f (- 10 \sqrt{30}) = 188 {(- \frac{20 \sqrt{30}}{{\sqrt{30}}^{2}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.4.6

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

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Step 15.2.1.4.6.1

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

$f (- 10 \sqrt{30}) = 188 {(- \frac{20 \sqrt{30}}{{(30^{\frac{1}{2}})}^{2}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.4.6.2

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

$f (- 10 \sqrt{30}) = 188 {(- \frac{20 \sqrt{30}}{30^{\frac{1}{2} \cdot 2}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.4.6.3

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

$f (- 10 \sqrt{30}) = 188 {(- \frac{20 \sqrt{30}}{30^{\frac{2}{2}}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.4.6.4

Cancel the common factor of $2$ .

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Step 15.2.1.4.6.4.1

Cancel the common factor.

$f (- 10 \sqrt{30}) = 188 {(- \frac{20 \sqrt{30}}{30^{\frac{2}{2}}} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.4.6.4.2

Rewrite the expression.

$f (- 10 \sqrt{30}) = 188 {(- \frac{20 \sqrt{30}}{30} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.4.6.5

Evaluate the exponent.

$f (- 10 \sqrt{30}) = 188 {(- \frac{20 \sqrt{30}}{30} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.5

Cancel the common factor of $20$ and $30$ .

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Step 15.2.1.5.1

Factor $10$ out of $20 \sqrt{30}$ .

$f (- 10 \sqrt{30}) = 188 {(- \frac{10 (2 \sqrt{30})}{30} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.5.2

Cancel the common factors.

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Step 15.2.1.5.2.1

Factor $10$ out of $30$ .

$f (- 10 \sqrt{30}) = 188 {(- \frac{10 (2 \sqrt{30})}{10 (3)} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.5.2.2

Cancel the common factor.

$f (- 10 \sqrt{30}) = 188 {(- \frac{10 (2 \sqrt{30})}{10 \cdot 3} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.5.2.3

Rewrite the expression.

$f (- 10 \sqrt{30}) = 188 {(- \frac{2 \sqrt{30}}{3} + \frac{- 10 \sqrt{30}}{15})}^{- 1}$

Step 15.2.1.6

Cancel the common factor of $- 10$ and $15$ .

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Step 15.2.1.6.1

Factor $5$ out of $- 10 \sqrt{30}$ .

$f (- 10 \sqrt{30}) = 188 {(- \frac{2 \sqrt{30}}{3} + \frac{5 (- 2 \sqrt{30})}{15})}^{- 1}$

Step 15.2.1.6.2

Cancel the common factors.

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Step 15.2.1.6.2.1

Factor $5$ out of $15$ .

$f (- 10 \sqrt{30}) = 188 {(- \frac{2 \sqrt{30}}{3} + \frac{5 (- 2 \sqrt{30})}{5 (3)})}^{- 1}$

Step 15.2.1.6.2.2

Cancel the common factor.

$f (- 10 \sqrt{30}) = 188 {(- \frac{2 \sqrt{30}}{3} + \frac{5 (- 2 \sqrt{30})}{5 \cdot 3})}^{- 1}$

Step 15.2.1.6.2.3

Rewrite the expression.

$f (- 10 \sqrt{30}) = 188 {(- \frac{2 \sqrt{30}}{3} + \frac{- 2 \sqrt{30}}{3})}^{- 1}$

Step 15.2.1.7

Move the negative in front of the fraction.

$f (- 10 \sqrt{30}) = 188 {(- \frac{2 \sqrt{30}}{3} - \frac{2 \sqrt{30}}{3})}^{- 1}$

Step 15.2.2

Simplify terms.

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Step 15.2.2.1

Combine the numerators over the common denominator.

$f (- 10 \sqrt{30}) = 188 {(\frac{- 2 \sqrt{30} - 2 \sqrt{30}}{3})}^{- 1}$

Step 15.2.2.2

Subtract $2 \sqrt{30}$ from $- 2 \sqrt{30}$ .

$f (- 10 \sqrt{30}) = 188 {(\frac{- 4 \sqrt{30}}{3})}^{- 1}$

Step 15.2.2.3

Move the negative in front of the fraction.

$f (- 10 \sqrt{30}) = 188 {(- \frac{4 \sqrt{30}}{3})}^{- 1}$

Step 15.2.3

Change the sign of the exponent by rewriting the base as its reciprocal.

$f (- 10 \sqrt{30}) = 188 (- \frac{3}{4 \sqrt{30}})$

Step 15.2.4

Cancel the common factor of $4$ .

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Step 15.2.4.1

Move the leading negative in $- \frac{3}{4 \sqrt{30}}$ into the numerator.

$f (- 10 \sqrt{30}) = 188 (\frac{- 3}{4 \sqrt{30}})$

Step 15.2.4.2

Factor $4$ out of $188$ .

$f (- 10 \sqrt{30}) = 4 (47) (\frac{- 3}{4 \sqrt{30}})$

Step 15.2.4.3

Factor $4$ out of $4 \sqrt{30}$ .

$f (- 10 \sqrt{30}) = 4 (47) (\frac{- 3}{4 (\sqrt{30})})$

Step 15.2.4.4

Cancel the common factor.

$f (- 10 \sqrt{30}) = 4 \cdot (47 (\frac{- 3}{4 \sqrt{30}}))$

Step 15.2.4.5

Rewrite the expression.

$f (- 10 \sqrt{30}) = 47 (\frac{- 3}{\sqrt{30}})$

Step 15.2.5

Combine $47$ and $\frac{- 3}{\sqrt{30}}$ .

$f (- 10 \sqrt{30}) = \frac{47 \cdot - 3}{\sqrt{30}}$

Step 15.2.6

Simplify the expression.

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Step 15.2.6.1

Multiply $47$ by $- 3$ .

$f (- 10 \sqrt{30}) = \frac{- 141}{\sqrt{30}}$

Step 15.2.6.2

Move the negative in front of the fraction.

$f (- 10 \sqrt{30}) = - \frac{141}{\sqrt{30}}$

Step 15.2.7

Multiply $\frac{141}{\sqrt{30}}$ by $\frac{\sqrt{30}}{\sqrt{30}}$ .

$f (- 10 \sqrt{30}) = - (\frac{141}{\sqrt{30}} \cdot \frac{\sqrt{30}}{\sqrt{30}})$

Step 15.2.8

Combine and simplify the denominator.

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Step 15.2.8.1

Multiply $\frac{141}{\sqrt{30}}$ by $\frac{\sqrt{30}}{\sqrt{30}}$ .

$f (- 10 \sqrt{30}) = - \frac{141 \sqrt{30}}{\sqrt{30} \sqrt{30}}$

Step 15.2.8.2

Raise $\sqrt{30}$ to the power of $1$ .

$f (- 10 \sqrt{30}) = - \frac{141 \sqrt{30}}{\sqrt{30} \sqrt{30}}$

Step 15.2.8.3

Raise $\sqrt{30}$ to the power of $1$ .

$f (- 10 \sqrt{30}) = - \frac{141 \sqrt{30}}{\sqrt{30} \sqrt{30}}$

Step 15.2.8.4

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

$f (- 10 \sqrt{30}) = - \frac{141 \sqrt{30}}{{\sqrt{30}}^{1 + 1}}$

Step 15.2.8.5

Add $1$ and $1$ .

$f (- 10 \sqrt{30}) = - \frac{141 \sqrt{30}}{{\sqrt{30}}^{2}}$

Step 15.2.8.6

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

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Step 15.2.8.6.1

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

$f (- 10 \sqrt{30}) = - \frac{141 \sqrt{30}}{{(30^{\frac{1}{2}})}^{2}}$

Step 15.2.8.6.2

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

$f (- 10 \sqrt{30}) = - \frac{141 \sqrt{30}}{30^{\frac{1}{2} \cdot 2}}$

Step 15.2.8.6.3

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

$f (- 10 \sqrt{30}) = - \frac{141 \sqrt{30}}{30^{\frac{2}{2}}}$

Step 15.2.8.6.4

Cancel the common factor of $2$ .

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Step 15.2.8.6.4.1

Cancel the common factor.

$f (- 10 \sqrt{30}) = - \frac{141 \sqrt{30}}{30^{\frac{2}{2}}}$

Step 15.2.8.6.4.2

Rewrite the expression.

$f (- 10 \sqrt{30}) = - \frac{141 \sqrt{30}}{30}$

Step 15.2.8.6.5

Evaluate the exponent.

$f (- 10 \sqrt{30}) = - \frac{141 \sqrt{30}}{30}$

Step 15.2.9

Cancel the common factor of $141$ and $30$ .

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Step 15.2.9.1

Factor $3$ out of $141 \sqrt{30}$ .

$f (- 10 \sqrt{30}) = - \frac{3 (47 \sqrt{30})}{30}$

Step 15.2.9.2

Cancel the common factors.

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Step 15.2.9.2.1

Factor $3$ out of $30$ .

$f (- 10 \sqrt{30}) = - \frac{3 (47 \sqrt{30})}{3 (10)}$

Step 15.2.9.2.2

Cancel the common factor.

$f (- 10 \sqrt{30}) = - \frac{3 (47 \sqrt{30})}{3 \cdot 10}$

Step 15.2.9.2.3

Rewrite the expression.

$f (- 10 \sqrt{30}) = - \frac{47 \sqrt{30}}{10}$

Step 15.2.10

The final answer is $- \frac{47 \sqrt{30}}{10}$ .

$y = - \frac{47 \sqrt{30}}{10}$

Step 16

These are the local extrema for $f (x) = 188 {(\frac{200}{x} + \frac{x}{15})}^{- 1}$ .

$(10 \sqrt{30}, \frac{47 \sqrt{30}}{10})$ is a local maxima

$(- 10 \sqrt{30}, - \frac{47 \sqrt{30}}{10})$ is a local minima

Step 17