Calculus Examples

$f (x) = \frac{x^{3}}{x^{2} + 5}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{3}$ and $g (x) = x^{2} + 5$ .

$f' (x) = \frac{(x^{2} + 5) \frac{d}{d x} (x^{3}) - x^{3} \frac{d}{d x} (x^{2} + 5)}{{(x^{2} + 5)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = \frac{(x^{2} + 5) (3 x^{2}) - x^{3} \frac{d}{d x} (x^{2} + 5)}{{(x^{2} + 5)}^{2}}$

Step 1.1.2.2

Move $3$ to the left of $x^{2} + 5$ .

$f' (x) = \frac{3 \cdot ((x^{2} + 5) x^{2}) - x^{3} \frac{d}{d x} (x^{2} + 5)}{{(x^{2} + 5)}^{2}}$

Step 1.1.2.3

By the Sum Rule, the derivative of $x^{2} + 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5]$ .

$f' (x) = \frac{3 (x^{2} + 5) x^{2} - x^{3} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (5))}{{(x^{2} + 5)}^{2}}$

Step 1.1.2.4

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

$f' (x) = \frac{3 (x^{2} + 5) x^{2} - x^{3} (2 x + \frac{d}{d x} (5))}{{(x^{2} + 5)}^{2}}$

Step 1.1.2.5

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$f' (x) = \frac{3 (x^{2} + 5) x^{2} - x^{3} (2 x + 0)}{{(x^{2} + 5)}^{2}}$

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $2 x$ and $0$ .

$f' (x) = \frac{3 (x^{2} + 5) x^{2} - x^{3} (2 x)}{{(x^{2} + 5)}^{2}}$

Step 1.1.2.6.2

Multiply $2$ by $- 1$ .

$f' (x) = \frac{3 (x^{2} + 5) x^{2} - 2 x^{3} x}{{(x^{2} + 5)}^{2}}$

Step 1.1.3

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

$f' (x) = \frac{3 (x^{2} + 5) x^{2} - 2 (x \cdot x^{3})}{{(x^{2} + 5)}^{2}}$

Step 1.1.4

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

$f' (x) = \frac{3 (x^{2} + 5) x^{2} - 2 x^{1 + 3}}{{(x^{2} + 5)}^{2}}$

Step 1.1.5

Add $1$ and $3$ .

$f' (x) = \frac{3 (x^{2} + 5) x^{2} - 2 x^{4}}{{(x^{2} + 5)}^{2}}$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Apply the distributive property.

$f' (x) = \frac{(3 x^{2} + 3 \cdot 5) x^{2} - 2 x^{4}}{{(x^{2} + 5)}^{2}}$

Step 1.1.6.2

Apply the distributive property.

$f' (x) = \frac{3 x^{2} x^{2} + 3 \cdot (5 x^{2}) - 2 x^{4}}{{(x^{2} + 5)}^{2}}$

Step 1.1.6.3

Simplify the numerator.

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Step 1.1.6.3.1

Simplify each term.

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Step 1.1.6.3.1.1

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

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Step 1.1.6.3.1.1.1

Move $x^{2}$ .

$f' (x) = \frac{3 (x^{2} x^{2}) + 3 \cdot (5 x^{2}) - 2 x^{4}}{{(x^{2} + 5)}^{2}}$

Step 1.1.6.3.1.1.2

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

$f' (x) = \frac{3 x^{2 + 2} + 3 \cdot (5 x^{2}) - 2 x^{4}}{{(x^{2} + 5)}^{2}}$

Step 1.1.6.3.1.1.3

Add $2$ and $2$ .

$f' (x) = \frac{3 x^{4} + 3 \cdot (5 x^{2}) - 2 x^{4}}{{(x^{2} + 5)}^{2}}$

Step 1.1.6.3.1.2

Multiply $3$ by $5$ .

$f' (x) = \frac{3 x^{4} + 15 x^{2} - 2 x^{4}}{{(x^{2} + 5)}^{2}}$

Step 1.1.6.3.2

Subtract $2 x^{4}$ from $3 x^{4}$ .

$f' (x) = \frac{x^{4} + 15 x^{2}}{{(x^{2} + 5)}^{2}}$

Step 1.1.6.4

Factor $x^{2}$ out of $x^{4} + 15 x^{2}$ .

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Step 1.1.6.4.1

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

$f' (x) = \frac{x^{2} x^{2} + 15 x^{2}}{{(x^{2} + 5)}^{2}}$

Step 1.1.6.4.2

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

$f' (x) = \frac{x^{2} x^{2} + x^{2} \cdot 15}{{(x^{2} + 5)}^{2}}$

Step 1.1.6.4.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot 15$ .

$f' (x) = \frac{x^{2} (x^{2} + 15)}{{(x^{2} + 5)}^{2}}$

Step 1.2

Find the second derivative.

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Step 1.2.1

$f'' (x) = \frac{{(x^{2} + 5)}^{2} \frac{d}{d x} (x^{2} (x^{2} + 15)) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{({(x^{2} + 5)}^{2})}^{2}}$

Step 1.2.2

Multiply the exponents in ${({(x^{2} + 5)}^{2})}^{2}$ .

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Step 1.2.2.1

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

$f'' (x) = \frac{{(x^{2} + 5)}^{2} \frac{d}{d x} (x^{2} (x^{2} + 15)) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{2 \cdot 2}}$

Step 1.2.2.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{{(x^{2} + 5)}^{2} \frac{d}{d x} (x^{2} (x^{2} + 15)) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 15$ .

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (x^{2} \frac{d}{d x} (x^{2} + 15) + (x^{2} + 15) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.4

Differentiate.

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Step 1.2.4.1

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

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (x^{2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (15)) + (x^{2} + 15) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.4.2

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

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (x^{2} (2 x + \frac{d}{d x} (15)) + (x^{2} + 15) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.4.3

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (x^{2} (2 x + 0) + (x^{2} + 15) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.4.4

Add $2 x$ and $0$ .

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (x^{2} (2 x) + (x^{2} + 15) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.5

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

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Step 1.2.5.1

Move $x$ .

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (x \cdot x^{2} \cdot 2 + (x^{2} + 15) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.5.2

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

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Step 1.2.5.2.1

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

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (x \cdot x^{2} \cdot 2 + (x^{2} + 15) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.5.2.2

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

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (x^{1 + 2} \cdot 2 + (x^{2} + 15) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.5.3

Add $1$ and $2$ .

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (x^{3} \cdot 2 + (x^{2} + 15) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.6

Differentiate using the Power Rule.

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Step 1.2.6.1

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

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (2 \cdot x^{3} + (x^{2} + 15) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.6.2

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

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (2 x^{3} + (x^{2} + 15) (2 x)) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.6.3

Move $2$ to the left of $x^{2} + 15$ .

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (2 x^{3} + 2 \cdot ((x^{2} + 15) x)) - x^{2} (x^{2} + 15) \frac{d}{d x} {(x^{2} + 5)}^{2}}{{(x^{2} + 5)}^{4}}$

Step 1.2.7

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 $g (x) = x^{2} + 5$ .

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Step 1.2.7.1

To apply the Chain Rule, set $u$ as $x^{2} + 5$ .

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (2 x^{3} + 2 (x^{2} + 15) x) - x^{2} (x^{2} + 15) (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} + 5))}{{(x^{2} + 5)}^{4}}$

Step 1.2.7.2

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

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (2 x^{3} + 2 (x^{2} + 15) x) - x^{2} (x^{2} + 15) (2 u \frac{d}{d x} (x^{2} + 5))}{{(x^{2} + 5)}^{4}}$

Step 1.2.7.3

Replace all occurrences of $u$ with $x^{2} + 5$ .

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (2 x^{3} + 2 (x^{2} + 15) x) - x^{2} (x^{2} + 15) (2 (x^{2} + 5) \frac{d}{d x} (x^{2} + 5))}{{(x^{2} + 5)}^{4}}$

Step 1.2.8

Simplify with factoring out.

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Step 1.2.8.1

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{{(x^{2} + 5)}^{2} (2 x^{3} + 2 (x^{2} + 15) x) - 2 x^{2} (x^{2} + 15) ((x^{2} + 5) \frac{d}{d x} (x^{2} + 5))}{{(x^{2} + 5)}^{4}}$

Step 1.2.8.2

Factor $x^{2} + 5$ out of ${(x^{2} + 5)}^{2} (2 x^{3} + 2 (x^{2} + 15) x) - 2 x^{2} (x^{2} + 15) ((x^{2} + 5) \frac{d}{d x} [x^{2} + 5])$ .

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Step 1.2.8.2.1

Factor $x^{2} + 5$ out of ${(x^{2} + 5)}^{2} (2 x^{3} + 2 (x^{2} + 15) x)$ .

$f'' (x) = \frac{(x^{2} + 5) ((x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x)) - 2 x^{2} (x^{2} + 15) ((x^{2} + 5) \frac{d}{d x} (x^{2} + 5))}{{(x^{2} + 5)}^{4}}$

Step 1.2.8.2.2

Factor $x^{2} + 5$ out of $- 2 x^{2} (x^{2} + 15) ((x^{2} + 5) \frac{d}{d x} [x^{2} + 5])$ .

$f'' (x) = \frac{(x^{2} + 5) ((x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x)) + (x^{2} + 5) (- 2 x^{2} (x^{2} + 15) (\frac{d}{d x} (x^{2} + 5)))}{{(x^{2} + 5)}^{4}}$

Step 1.2.8.2.3

Factor $x^{2} + 5$ out of $(x^{2} + 5) ((x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x)) + (x^{2} + 5) (- 2 x^{2} (x^{2} + 15) (\frac{d}{d x} [x^{2} + 5]))$ .

$f'' (x) = \frac{(x^{2} + 5) ((x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x) - 2 x^{2} (x^{2} + 15) (\frac{d}{d x} (x^{2} + 5)))}{{(x^{2} + 5)}^{4}}$

Step 1.2.9

Cancel the common factors.

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Step 1.2.9.1

Factor $x^{2} + 5$ out of ${(x^{2} + 5)}^{4}$ .

$f'' (x) = \frac{(x^{2} + 5) ((x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x) - 2 x^{2} (x^{2} + 15) (\frac{d}{d x} (x^{2} + 5)))}{(x^{2} + 5) {(x^{2} + 5)}^{3}}$

Step 1.2.9.2

Cancel the common factor.

$f'' (x) = \frac{(x^{2} + 5) ((x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x) - 2 x^{2} (x^{2} + 15) (\frac{d}{d x} (x^{2} + 5)))}{(x^{2} + 5) {(x^{2} + 5)}^{3}}$

Step 1.2.9.3

Rewrite the expression.

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x) - 2 x^{2} (x^{2} + 15) (\frac{d}{d x} (x^{2} + 5))}{{(x^{2} + 5)}^{3}}$

Step 1.2.10

By the Sum Rule, the derivative of $x^{2} + 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5]$ .

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x) - 2 x^{2} (x^{2} + 15) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (5))}{{(x^{2} + 5)}^{3}}$

Step 1.2.11

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

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x) - 2 x^{2} (x^{2} + 15) (2 x + \frac{d}{d x} (5))}{{(x^{2} + 5)}^{3}}$

Step 1.2.12

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x) - 2 x^{2} (x^{2} + 15) (2 x + 0)}{{(x^{2} + 5)}^{3}}$

Step 1.2.13

Simplify the expression.

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Step 1.2.13.1

Add $2 x$ and $0$ .

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x) - 2 x^{2} (x^{2} + 15) (2 x)}{{(x^{2} + 5)}^{3}}$

Step 1.2.13.2

Multiply $2$ by $- 2$ .

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x) - 4 x^{2} (x^{2} + 15) x}{{(x^{2} + 5)}^{3}}$

Step 1.2.14

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

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x) - 4 (x \cdot x^{2}) (x^{2} + 15)}{{(x^{2} + 5)}^{3}}$

Step 1.2.15

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

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x) - 4 x^{1 + 2} (x^{2} + 15)}{{(x^{2} + 5)}^{3}}$

Step 1.2.16

Add $1$ and $2$ .

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 (x^{2} + 15) x) - 4 x^{3} (x^{2} + 15)}{{(x^{2} + 5)}^{3}}$

Step 1.2.17

Simplify.

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Step 1.2.17.1

Apply the distributive property.

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + (2 x^{2} + 2 \cdot 15) x) - 4 x^{3} (x^{2} + 15)}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.2

Apply the distributive property.

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 x^{2} x + 2 \cdot (15 x)) - 4 x^{3} (x^{2} + 15)}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.3

Apply the distributive property.

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 x^{2} x + 2 \cdot (15 x)) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4

Simplify the numerator.

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Step 1.2.17.4.1

Simplify each term.

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Step 1.2.17.4.1.1

Simplify each term.

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Step 1.2.17.4.1.1.1

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

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Step 1.2.17.4.1.1.1.1

Move $x$ .

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 (x \cdot x^{2}) + 2 \cdot (15 x)) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.1.1.2

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

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Step 1.2.17.4.1.1.1.2.1

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

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 (x \cdot x^{2}) + 2 \cdot (15 x)) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.1.1.2.2

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

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 x^{1 + 2} + 2 \cdot (15 x)) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.1.1.3

Add $1$ and $2$ .

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 x^{3} + 2 \cdot (15 x)) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.1.2

Multiply $2$ by $15$ .

$f'' (x) = \frac{(x^{2} + 5) (2 x^{3} + 2 x^{3} + 30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.2

Add $2 x^{3}$ and $2 x^{3}$ .

$f'' (x) = \frac{(x^{2} + 5) (4 x^{3} + 30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.3

Expand $(x^{2} + 5) (4 x^{3} + 30 x)$ using the FOIL Method.

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Step 1.2.17.4.1.3.1

Apply the distributive property.

$f'' (x) = \frac{x^{2} (4 x^{3} + 30 x) + 5 (4 x^{3} + 30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.3.2

Apply the distributive property.

$f'' (x) = \frac{x^{2} (4 x^{3}) + x^{2} (30 x) + 5 (4 x^{3} + 30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.3.3

Apply the distributive property.

$f'' (x) = \frac{x^{2} (4 x^{3}) + x^{2} (30 x) + 5 (4 x^{3}) + 5 (30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.4

Simplify and combine like terms.

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Step 1.2.17.4.1.4.1

Simplify each term.

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Step 1.2.17.4.1.4.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{2} x^{3} + x^{2} (30 x) + 5 (4 x^{3}) + 5 (30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.4.1.2

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 1.2.17.4.1.4.1.2.1

Move $x^{3}$ .

$f'' (x) = \frac{4 (x^{3} x^{2}) + x^{2} (30 x) + 5 (4 x^{3}) + 5 (30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.4.1.2.2

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

$f'' (x) = \frac{4 x^{3 + 2} + x^{2} (30 x) + 5 (4 x^{3}) + 5 (30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.4.1.2.3

Add $3$ and $2$ .

$f'' (x) = \frac{4 x^{5} + x^{2} (30 x) + 5 (4 x^{3}) + 5 (30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.4.1.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{5} + 30 x^{2} x + 5 (4 x^{3}) + 5 (30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.4.1.4

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

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Step 1.2.17.4.1.4.1.4.1

Move $x$ .

$f'' (x) = \frac{4 x^{5} + 30 (x \cdot x^{2}) + 5 (4 x^{3}) + 5 (30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.4.1.4.2

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

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Step 1.2.17.4.1.4.1.4.2.1

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

$f'' (x) = \frac{4 x^{5} + 30 (x \cdot x^{2}) + 5 (4 x^{3}) + 5 (30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.4.1.4.2.2

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

$f'' (x) = \frac{4 x^{5} + 30 x^{1 + 2} + 5 (4 x^{3}) + 5 (30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.4.1.4.3

Add $1$ and $2$ .

$f'' (x) = \frac{4 x^{5} + 30 x^{3} + 5 (4 x^{3}) + 5 (30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.4.1.5

Multiply $4$ by $5$ .

$f'' (x) = \frac{4 x^{5} + 30 x^{3} + 20 x^{3} + 5 (30 x) - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.4.1.6

Multiply $30$ by $5$ .

$f'' (x) = \frac{4 x^{5} + 30 x^{3} + 20 x^{3} + 150 x - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.4.2

Add $30 x^{3}$ and $20 x^{3}$ .

$f'' (x) = \frac{4 x^{5} + 50 x^{3} + 150 x - 4 x^{3} x^{2} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.5

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 1.2.17.4.1.5.1

Move $x^{2}$ .

$f'' (x) = \frac{4 x^{5} + 50 x^{3} + 150 x - 4 (x^{2} x^{3}) - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.5.2

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

$f'' (x) = \frac{4 x^{5} + 50 x^{3} + 150 x - 4 x^{2 + 3} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.5.3

Add $2$ and $3$ .

$f'' (x) = \frac{4 x^{5} + 50 x^{3} + 150 x - 4 x^{5} - 4 x^{3} \cdot 15}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.1.6

Multiply $15$ by $- 4$ .

$f'' (x) = \frac{4 x^{5} + 50 x^{3} + 150 x - 4 x^{5} - 60 x^{3}}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.2

Combine the opposite terms in $4 x^{5} + 50 x^{3} + 150 x - 4 x^{5} - 60 x^{3}$ .

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Step 1.2.17.4.2.1

Subtract $4 x^{5}$ from $4 x^{5}$ .

$f'' (x) = \frac{50 x^{3} + 150 x + 0 - 60 x^{3}}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.2.2

Add $50 x^{3} + 150 x$ and $0$ .

$f'' (x) = \frac{50 x^{3} + 150 x - 60 x^{3}}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.4.3

Subtract $60 x^{3}$ from $50 x^{3}$ .

$f'' (x) = \frac{- 10 x^{3} + 150 x}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.5

Factor $10 x$ out of $- 10 x^{3} + 150 x$ .

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Step 1.2.17.5.1

Factor $10 x$ out of $- 10 x^{3}$ .

$f'' (x) = \frac{10 x (- x^{2}) + 150 x}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.5.2

Factor $10 x$ out of $150 x$ .

$f'' (x) = \frac{10 x (- x^{2}) + 10 x (15)}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.5.3

Factor $10 x$ out of $10 x (- x^{2}) + 10 x (15)$ .

$f'' (x) = \frac{10 x (- x^{2} + 15)}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.6

Factor $- 1$ out of $- x^{2}$ .

$f'' (x) = \frac{10 x (- (x^{2}) + 15)}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.7

Rewrite $15$ as $- 1 (- 15)$ .

$f'' (x) = \frac{10 x (- (x^{2}) - 1 \cdot - 15)}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.8

Factor $- 1$ out of $- (x^{2}) - 1 (- 15)$ .

$f'' (x) = \frac{10 x (- (x^{2} - 15))}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.9

Rewrite $- (x^{2} - 15)$ as $- 1 (x^{2} - 15)$ .

$f'' (x) = \frac{10 x (- 1 (x^{2} - 15))}{{(x^{2} + 5)}^{3}}$

Step 1.2.17.10

Move the negative in front of the fraction.

$f'' (x) = - \frac{(10 x) (x^{2} - 15)}{{(x^{2} + 5)}^{3}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{(10 x) (x^{2} - 15)}{{(x^{2} + 5)}^{3}}$ .

$- \frac{(10 x) (x^{2} - 15)}{{(x^{2} + 5)}^{3}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $- \frac{(10 x) (x^{2} - 15)}{{(x^{2} + 5)}^{3}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- \frac{(10 x) (x^{2} - 15)}{{(x^{2} + 5)}^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$10 x (x^{2} - 15) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

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

$x = 0$

$x^{2} - 15 = 0$

Step 2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.3.3

Set $x^{2} - 15$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x^{2} - 15$ equal to $0$ .

$x^{2} - 15 = 0$

Step 2.3.3.2

Solve $x^{2} - 15 = 0$ for $x$ .

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Step 2.3.3.2.1

Add $15$ to both sides of the equation.

$x^{2} = 15$

Step 2.3.3.2.2

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

$x = \pm \sqrt{15}$

Step 2.3.3.2.3

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

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Step 2.3.3.2.3.1

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

$x = \sqrt{15}$

Step 2.3.3.2.3.2

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

$x = - \sqrt{15}$

Step 2.3.3.2.3.3

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

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

Step 2.3.4

The final solution is all the values that make $10 x (x^{2} - 15) = 0$ true.

$x = 0, \sqrt{15}, - \sqrt{15}$

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $0$ in $f (x) = \frac{x^{3}}{x^{2} + 5}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{{(0)}^{3}}{{(0)}^{2} + 5}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

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

$f (0) = \frac{0}{0^{2} + 5}$

Step 3.1.2.2

Simplify the denominator.

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Step 3.1.2.2.1

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

$f (0) = \frac{0}{0 + 5}$

Step 3.1.2.2.2

Add $0$ and $5$ .

$f (0) = \frac{0}{5}$

Step 3.1.2.3

Divide $0$ by $5$ .

$f (0) = 0$

Step 3.1.2.4

The final answer is $0$ .

$0$

Step 3.2

The point found by substituting $0$ in $f (x) = \frac{x^{3}}{x^{2} + 5}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 3.3

Substitute $\sqrt{15}$ in $f (x) = \frac{x^{3}}{x^{2} + 5}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $\sqrt{15}$ in the expression.

$f (\sqrt{15}) = \frac{{(\sqrt{15})}^{3}}{{(\sqrt{15})}^{2} + 5}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify the numerator.

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Step 3.3.2.1.1

Rewrite ${\sqrt{15}}^{3}$ as $\sqrt{15^{3}}$ .

$f (\sqrt{15}) = \frac{\sqrt{15^{3}}}{{\sqrt{15}}^{2} + 5}$

Step 3.3.2.1.2

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

$f (\sqrt{15}) = \frac{\sqrt{3375}}{{\sqrt{15}}^{2} + 5}$

Step 3.3.2.1.3

Rewrite $3375$ as $15^{2} \cdot 15$ .

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Step 3.3.2.1.3.1

Factor $225$ out of $3375$ .

$f (\sqrt{15}) = \frac{\sqrt{225 (15)}}{{\sqrt{15}}^{2} + 5}$

Step 3.3.2.1.3.2

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

$f (\sqrt{15}) = \frac{\sqrt{15^{2} \cdot 15}}{{\sqrt{15}}^{2} + 5}$

Step 3.3.2.1.4

Pull terms out from under the radical.

$f (\sqrt{15}) = \frac{15 \sqrt{15}}{{\sqrt{15}}^{2} + 5}$

Step 3.3.2.2

Simplify the denominator.

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Step 3.3.2.2.1

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

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Step 3.3.2.2.1.1

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

$f (\sqrt{15}) = \frac{15 \sqrt{15}}{{(15^{\frac{1}{2}})}^{2} + 5}$

Step 3.3.2.2.1.2

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

$f (\sqrt{15}) = \frac{15 \sqrt{15}}{15^{\frac{1}{2} \cdot 2} + 5}$

Step 3.3.2.2.1.3

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

$f (\sqrt{15}) = \frac{15 \sqrt{15}}{15^{\frac{2}{2}} + 5}$

Step 3.3.2.2.1.4

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.4.1

Cancel the common factor.

$f (\sqrt{15}) = \frac{15 \sqrt{15}}{15^{\frac{2}{2}} + 5}$

Step 3.3.2.2.1.4.2

Rewrite the expression.

$f (\sqrt{15}) = \frac{15 \sqrt{15}}{15 + 5}$

Step 3.3.2.2.1.5

Evaluate the exponent.

$f (\sqrt{15}) = \frac{15 \sqrt{15}}{15 + 5}$

Step 3.3.2.2.2

Add $15$ and $5$ .

$f (\sqrt{15}) = \frac{15 \sqrt{15}}{20}$

Step 3.3.2.3

Cancel the common factor of $15$ and $20$ .

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Step 3.3.2.3.1

Factor $5$ out of $15 \sqrt{15}$ .

$f (\sqrt{15}) = \frac{5 (3 \sqrt{15})}{20}$

Step 3.3.2.3.2

Cancel the common factors.

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Step 3.3.2.3.2.1

Factor $5$ out of $20$ .

$f (\sqrt{15}) = \frac{5 (3 \sqrt{15})}{5 (4)}$

Step 3.3.2.3.2.2

Cancel the common factor.

$f (\sqrt{15}) = \frac{5 (3 \sqrt{15})}{5 \cdot 4}$

Step 3.3.2.3.2.3

Rewrite the expression.

$f (\sqrt{15}) = \frac{3 \sqrt{15}}{4}$

Step 3.3.2.4

The final answer is $\frac{3 \sqrt{15}}{4}$ .

$\frac{3 \sqrt{15}}{4}$

Step 3.4

The point found by substituting $\sqrt{15}$ in $f (x) = \frac{x^{3}}{x^{2} + 5}$ is $(\sqrt{15}, \frac{3 \sqrt{15}}{4})$ . This point can be an inflection point.

$(\sqrt{15}, \frac{3 \sqrt{15}}{4})$

Step 3.5

Substitute $- \sqrt{15}$ in $f (x) = \frac{x^{3}}{x^{2} + 5}$ to find the value of $y$ .

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Step 3.5.1

Replace the variable $x$ with $- \sqrt{15}$ in the expression.

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

Step 3.5.2

Simplify the result.

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Step 3.5.2.1

Simplify the numerator.

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Step 3.5.2.1.1

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

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

Step 3.5.2.1.2

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

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

Step 3.5.2.1.3

Rewrite ${\sqrt{15}}^{3}$ as $\sqrt{15^{3}}$ .

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

Step 3.5.2.1.4

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

$f (- \sqrt{15}) = \frac{- \sqrt{3375}}{{(- \sqrt{15})}^{2} + 5}$

Step 3.5.2.1.5

Rewrite $3375$ as $15^{2} \cdot 15$ .

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Step 3.5.2.1.5.1

Factor $225$ out of $3375$ .

$f (- \sqrt{15}) = \frac{- \sqrt{225 (15)}}{{(- \sqrt{15})}^{2} + 5}$

Step 3.5.2.1.5.2

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

$f (- \sqrt{15}) = \frac{- \sqrt{15^{2} \cdot 15}}{{(- \sqrt{15})}^{2} + 5}$

Step 3.5.2.1.6

Pull terms out from under the radical.

$f (- \sqrt{15}) = \frac{- 1 \cdot (15 \sqrt{15})}{{(- \sqrt{15})}^{2} + 5}$

Step 3.5.2.1.7

Multiply $- 1$ by $15$ .

$f (- \sqrt{15}) = \frac{- 15 \sqrt{15}}{{(- \sqrt{15})}^{2} + 5}$

Step 3.5.2.2

Simplify the denominator.

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Step 3.5.2.2.1

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

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

Step 3.5.2.2.2

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

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

Step 3.5.2.2.3

Multiply ${\sqrt{15}}^{2}$ by $1$ .

$f (- \sqrt{15}) = \frac{- 15 \sqrt{15}}{{\sqrt{15}}^{2} + 5}$

Step 3.5.2.2.4

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

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Step 3.5.2.2.4.1

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

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

Step 3.5.2.2.4.2

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

$f (- \sqrt{15}) = \frac{- 15 \sqrt{15}}{15^{\frac{1}{2} \cdot 2} + 5}$

Step 3.5.2.2.4.3

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

$f (- \sqrt{15}) = \frac{- 15 \sqrt{15}}{15^{\frac{2}{2}} + 5}$

Step 3.5.2.2.4.4

Cancel the common factor of $2$ .

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Step 3.5.2.2.4.4.1

Cancel the common factor.

$f (- \sqrt{15}) = \frac{- 15 \sqrt{15}}{15^{\frac{2}{2}} + 5}$

Step 3.5.2.2.4.4.2

Rewrite the expression.

$f (- \sqrt{15}) = \frac{- 15 \sqrt{15}}{15 + 5}$

Step 3.5.2.2.4.5

Evaluate the exponent.

$f (- \sqrt{15}) = \frac{- 15 \sqrt{15}}{15 + 5}$

Step 3.5.2.2.5

Add $15$ and $5$ .

$f (- \sqrt{15}) = \frac{- 15 \sqrt{15}}{20}$

Step 3.5.2.3

Reduce the expression by cancelling the common factors.

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Step 3.5.2.3.1

Cancel the common factor of $- 15$ and $20$ .

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Step 3.5.2.3.1.1

Factor $5$ out of $- 15 \sqrt{15}$ .

$f (- \sqrt{15}) = \frac{5 (- 3 \sqrt{15})}{20}$

Step 3.5.2.3.1.2

Cancel the common factors.

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Step 3.5.2.3.1.2.1

Factor $5$ out of $20$ .

$f (- \sqrt{15}) = \frac{5 (- 3 \sqrt{15})}{5 (4)}$

Step 3.5.2.3.1.2.2

Cancel the common factor.

$f (- \sqrt{15}) = \frac{5 (- 3 \sqrt{15})}{5 \cdot 4}$

Step 3.5.2.3.1.2.3

Rewrite the expression.

$f (- \sqrt{15}) = \frac{- 3 \sqrt{15}}{4}$

Step 3.5.2.3.2

Move the negative in front of the fraction.

$f (- \sqrt{15}) = - \frac{3 \sqrt{15}}{4}$

Step 3.5.2.4

The final answer is $- \frac{3 \sqrt{15}}{4}$ .

$- \frac{3 \sqrt{15}}{4}$

Step 3.6

The point found by substituting $- \sqrt{15}$ in $f (x) = \frac{x^{3}}{x^{2} + 5}$ is $(- \sqrt{15}, - \frac{3 \sqrt{15}}{4})$ . This point can be an inflection point.

$(- \sqrt{15}, - \frac{3 \sqrt{15}}{4})$

Step 3.7

Determine the points that could be inflection points.

$(0, 0), (\sqrt{15}, \frac{3 \sqrt{15}}{4}), (- \sqrt{15}, - \frac{3 \sqrt{15}}{4})$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - \sqrt{15}) \cup (- \sqrt{15}, 0) \cup (0, \sqrt{15}) \cup (\sqrt{15}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \sqrt{15})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 3.97298334$ in the expression.

$f'' (- 3.97298334) = - \frac{(10 (- 3.97298334)) ({(- 3.97298334)}^{2} - 15)}{{({(- 3.97298334)}^{2} + 5)}^{3}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Multiply $10$ by $- 3.97298334$ .

$f'' (- 3.97298334) = - \frac{- 39.72983346 ({(- 3.97298334)}^{2} - 15)}{{({(- 3.97298334)}^{2} + 5)}^{3}}$

Step 5.2.1.2

Multiply $- 39.72983346$ by ${(- 3.97298334)}^{2} - 15$ .

$f'' (- 3.97298334) = - \frac{- 31.171895}{{({(- 3.97298334)}^{2} + 5)}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Raise $- 3.97298334$ to the power of $2$ .

$f'' (- 3.97298334) = - \frac{- 31.171895}{{(15.78459666 + 5)}^{3}}$

Step 5.2.2.2

Add $15.78459666$ and $5$ .

$f'' (- 3.97298334) = - \frac{- 31.171895}{{20.78459666}^{3}}$

Step 5.2.2.3

Raise $20.78459666$ to the power of $3$ .

$f'' (- 3.97298334) = - \frac{- 31.171895}{8978.93451047}$

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Divide $- 31.171895$ by $8978.93451047$ .

$f'' (- 3.97298334) = 0.00347166$

Step 5.2.3.2

Multiply $- 1$ by $- 0.00347166$ .

$f'' (- 3.97298334) = 0.00347166$

Step 5.2.4

The final answer is $0.00347166$ .

$0.00347166$

Step 5.3

At $- 3.97298334$ , the second derivative is $0.00347166$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - \sqrt{15})$ .

Increasing on $(- \infty, - \sqrt{15})$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- \sqrt{15}, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1.93649167$ in the expression.

$f'' (- 1.93649167) = - \frac{(10 (- 1.93649167)) ({(- 1.93649167)}^{2} - 15)}{{({(- 1.93649167)}^{2} + 5)}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Multiply $10$ by $- 1.93649167$ .

$f'' (- 1.93649167) = - \frac{- 19.36491673 ({(- 1.93649167)}^{2} - 15)}{{({(- 1.93649167)}^{2} + 5)}^{3}}$

Step 6.2.1.2

Multiply $- 19.36491673$ by ${(- 1.93649167)}^{2} - 15$ .

$f'' (- 1.93649167) = - \frac{217.85531322}{{({(- 1.93649167)}^{2} + 5)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Raise $- 1.93649167$ to the power of $2$ .

$f'' (- 1.93649167) = - \frac{217.85531322}{{(3.75 + 5)}^{3}}$

Step 6.2.2.2

Add $3.75$ and $5$ .

$f'' (- 1.93649167) = - \frac{217.85531322}{{8.75}^{3}}$

Step 6.2.2.3

Raise $8.75$ to the power of $3$ .

$f'' (- 1.93649167) = - \frac{217.85531322}{669.921875}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Divide $217.85531322$ by $669.921875$ .

$f'' (- 1.93649167) = - 1 \cdot 0.3251951$

Step 6.2.3.2

Multiply $- 1$ by $0.3251951$ .

$f'' (- 1.93649167) = - 0.3251951$

Step 6.2.4

The final answer is $- 0.3251951$ .

$- 0.3251951$

Step 6.3

At $- 1.93649167$ , the second derivative is $- 0.3251951$ . Since this is negative, the second derivative is decreasing on the interval $(- \sqrt{15}, 0)$

Decreasing on $(- \sqrt{15}, 0)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(0, \sqrt{15})$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $1.93649167$ in the expression.

$f'' (1.93649167) = - \frac{(10 (1.93649167)) ({(1.93649167)}^{2} - 15)}{{({(1.93649167)}^{2} + 5)}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Multiply $10$ by $1.93649167$ .

$f'' (1.93649167) = - \frac{19.36491673 ({1.93649167}^{2} - 15)}{{({1.93649167}^{2} + 5)}^{3}}$

Step 7.2.1.2

Multiply $19.36491673$ by ${1.93649167}^{2} - 15$ .

$f'' (1.93649167) = - \frac{- 217.85531322}{{({1.93649167}^{2} + 5)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Raise $1.93649167$ to the power of $2$ .

$f'' (1.93649167) = - \frac{- 217.85531322}{{(3.75 + 5)}^{3}}$

Step 7.2.2.2

Add $3.75$ and $5$ .

$f'' (1.93649167) = - \frac{- 217.85531322}{{8.75}^{3}}$

Step 7.2.2.3

Raise $8.75$ to the power of $3$ .

$f'' (1.93649167) = - \frac{- 217.85531322}{669.921875}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Divide $- 217.85531322$ by $669.921875$ .

$f'' (1.93649167) = 0.3251951$

Step 7.2.3.2

Multiply $- 1$ by $- 0.3251951$ .

$f'' (1.93649167) = 0.3251951$

Step 7.2.4

The final answer is $0.3251951$ .

$0.3251951$

Step 7.3

At $1.93649167$ , the second derivative is $0.3251951$ . Since this is positive, the second derivative is increasing on the interval $(0, \sqrt{15})$ .

Increasing on $(0, \sqrt{15})$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(\sqrt{15}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $3.97298334$ in the expression.

$f'' (3.97298334) = - \frac{(10 (3.97298334)) ({(3.97298334)}^{2} - 15)}{{({(3.97298334)}^{2} + 5)}^{3}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

Multiply $10$ by $3.97298334$ .

$f'' (3.97298334) = - \frac{39.72983346 ({3.97298334}^{2} - 15)}{{({3.97298334}^{2} + 5)}^{3}}$

Step 8.2.1.2

Multiply $39.72983346$ by ${3.97298334}^{2} - 15$ .

$f'' (3.97298334) = - \frac{31.171895}{{({3.97298334}^{2} + 5)}^{3}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Raise $3.97298334$ to the power of $2$ .

$f'' (3.97298334) = - \frac{31.171895}{{(15.78459666 + 5)}^{3}}$

Step 8.2.2.2

Add $15.78459666$ and $5$ .

$f'' (3.97298334) = - \frac{31.171895}{{20.78459666}^{3}}$

Step 8.2.2.3

Raise $20.78459666$ to the power of $3$ .

$f'' (3.97298334) = - \frac{31.171895}{8978.93451047}$

Step 8.2.3

Simplify the expression.

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Step 8.2.3.1

Divide $31.171895$ by $8978.93451047$ .

$f'' (3.97298334) = - 1 \cdot 0.00347166$

Step 8.2.3.2

Multiply $- 1$ by $0.00347166$ .

$f'' (3.97298334) = - 0.00347166$

Step 8.2.4

The final answer is $- 0.00347166$ .

$- 0.00347166$

Step 8.3

At $3.97298334$ , the second derivative is $- 0.00347166$ . Since this is negative, the second derivative is decreasing on the interval $(\sqrt{15}, \infty)$

Decreasing on $(\sqrt{15}, \infty)$ since $f'' (x) < 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(- \sqrt{15}, - \frac{3 \sqrt{15}}{4}), (0, 0), (\sqrt{15}, \frac{3 \sqrt{15}}{4})$ .

$(- \sqrt{15}, - \frac{3 \sqrt{15}}{4}), (0, 0), (\sqrt{15}, \frac{3 \sqrt{15}}{4})$

Step 10