Calculus Examples

$y = \frac{x}{x^{2} + 25}$

Step 1

Write $y = \frac{x}{x^{2} + 25}$ as a function.

$f (x) = \frac{x}{x^{2} + 25}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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 $f (x) = x$ and $g (x) = x^{2} + 25$ .

$f' (x) = \frac{(x^{2} + 25) \frac{d}{d x} (x) - x \frac{d}{d x} (x^{2} + 25)}{{(x^{2} + 25)}^{2}}$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

$f' (x) = \frac{(x^{2} + 25) \cdot 1 - x \frac{d}{d x} (x^{2} + 25)}{{(x^{2} + 25)}^{2}}$

Step 2.1.2.2

Multiply $x^{2} + 25$ by $1$ .

$f' (x) = \frac{x^{2} + 25 - x \frac{d}{d x} (x^{2} + 25)}{{(x^{2} + 25)}^{2}}$

Step 2.1.2.3

By the Sum Rule, the derivative of $x^{2} + 25$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [25]$ .

$f' (x) = \frac{x^{2} + 25 - x (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (25))}{{(x^{2} + 25)}^{2}}$

Step 2.1.2.4

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

$f' (x) = \frac{x^{2} + 25 - x (2 x + \frac{d}{d x} (25))}{{(x^{2} + 25)}^{2}}$

Step 2.1.2.5

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$f' (x) = \frac{x^{2} + 25 - x (2 x + 0)}{{(x^{2} + 25)}^{2}}$

Step 2.1.2.6

Simplify the expression.

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Step 2.1.2.6.1

Add $2 x$ and $0$ .

$f' (x) = \frac{x^{2} + 25 - x (2 x)}{{(x^{2} + 25)}^{2}}$

Step 2.1.2.6.2

Multiply $2$ by $- 1$ .

$f' (x) = \frac{x^{2} + 25 - 2 x \cdot x}{{(x^{2} + 25)}^{2}}$

Step 2.1.3

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

$f' (x) = \frac{x^{2} + 25 - 2 (x \cdot x)}{{(x^{2} + 25)}^{2}}$

Step 2.1.4

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

$f' (x) = \frac{x^{2} + 25 - 2 (x \cdot x)}{{(x^{2} + 25)}^{2}}$

Step 2.1.5

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

$f' (x) = \frac{x^{2} + 25 - 2 x^{1 + 1}}{{(x^{2} + 25)}^{2}}$

Step 2.1.6

Add $1$ and $1$ .

$f' (x) = \frac{x^{2} + 25 - 2 x^{2}}{{(x^{2} + 25)}^{2}}$

Step 2.1.7

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

$f' (x) = \frac{- x^{2} + 25}{{(x^{2} + 25)}^{2}}$

Step 2.2

Find the second derivative.

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Step 2.2.1

$f'' (x) = \frac{{(x^{2} + 25)}^{2} \frac{d}{d x} (- x^{2} + 25) - (- x^{2} + 25) \frac{d}{d x} {(x^{2} + 25)}^{2}}{{({(x^{2} + 25)}^{2})}^{2}}$

Step 2.2.2

Differentiate.

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Step 2.2.2.1

Multiply the exponents in ${({(x^{2} + 25)}^{2})}^{2}$ .

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Step 2.2.2.1.1

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

$f'' (x) = \frac{{(x^{2} + 25)}^{2} \frac{d}{d x} (- x^{2} + 25) - (- x^{2} + 25) \frac{d}{d x} {(x^{2} + 25)}^{2}}{{(x^{2} + 25)}^{2 \cdot 2}}$

Step 2.2.2.1.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} \frac{d}{d x} (- x^{2} + 25) - (- x^{2} + 25) \frac{d}{d x} {(x^{2} + 25)}^{2}}{{(x^{2} + 25)}^{4}}$

Step 2.2.2.2

By the Sum Rule, the derivative of $- x^{2} + 25$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [25]$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (\frac{d}{d x} (- x^{2}) + \frac{d}{d x} (25)) - (- x^{2} + 25) \frac{d}{d x} {(x^{2} + 25)}^{2}}{{(x^{2} + 25)}^{4}}$

Step 2.2.2.3

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

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- \frac{d}{d x} x^{2} + \frac{d}{d x} (25)) - (- x^{2} + 25) \frac{d}{d x} {(x^{2} + 25)}^{2}}{{(x^{2} + 25)}^{4}}$

Step 2.2.2.4

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

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- (2 x) + \frac{d}{d x} (25)) - (- x^{2} + 25) \frac{d}{d x} {(x^{2} + 25)}^{2}}{{(x^{2} + 25)}^{4}}$

Step 2.2.2.5

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x + \frac{d}{d x} (25)) - (- x^{2} + 25) \frac{d}{d x} {(x^{2} + 25)}^{2}}{{(x^{2} + 25)}^{4}}$

Step 2.2.2.6

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x + 0) - (- x^{2} + 25) \frac{d}{d x} {(x^{2} + 25)}^{2}}{{(x^{2} + 25)}^{4}}$

Step 2.2.2.7

Add $- 2 x$ and $0$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) - (- x^{2} + 25) \frac{d}{d x} {(x^{2} + 25)}^{2}}{{(x^{2} + 25)}^{4}}$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 25$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u$ as $x^{2} + 25$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) - (- x^{2} + 25) (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} + 25))}{{(x^{2} + 25)}^{4}}$

Step 2.2.3.2

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

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) - (- x^{2} + 25) (2 u \frac{d}{d x} (x^{2} + 25))}{{(x^{2} + 25)}^{4}}$

Step 2.2.3.3

Replace all occurrences of $u$ with $x^{2} + 25$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) - (- x^{2} + 25) (2 (x^{2} + 25) \frac{d}{d x} (x^{2} + 25))}{{(x^{2} + 25)}^{4}}$

Step 2.2.4

Differentiate.

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Step 2.2.4.1

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) - 2 (- x^{2} + 25) ((x^{2} + 25) \frac{d}{d x} (x^{2} + 25))}{{(x^{2} + 25)}^{4}}$

Step 2.2.4.2

By the Sum Rule, the derivative of $x^{2} + 25$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [25]$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) - 2 (- x^{2} + 25) ((x^{2} + 25) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (25)))}{{(x^{2} + 25)}^{4}}$

Step 2.2.4.3

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

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) - 2 (- x^{2} + 25) ((x^{2} + 25) (2 x + \frac{d}{d x} (25)))}{{(x^{2} + 25)}^{4}}$

Step 2.2.4.4

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) - 2 (- x^{2} + 25) ((x^{2} + 25) (2 x + 0))}{{(x^{2} + 25)}^{4}}$

Step 2.2.4.5

Simplify the expression.

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Step 2.2.4.5.1

Add $2 x$ and $0$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) - 2 (- x^{2} + 25) ((x^{2} + 25) (2 x))}{{(x^{2} + 25)}^{4}}$

Step 2.2.4.5.2

Move $2$ to the left of $x^{2} + 25$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) - 2 (- x^{2} + 25) (2 \cdot ((x^{2} + 25) x))}{{(x^{2} + 25)}^{4}}$

Step 2.2.4.5.3

Multiply $2$ by $- 2$ .

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) - 4 (- x^{2} + 25) ((x^{2} + 25) x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5

Simplify.

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Step 2.2.5.1

Apply the distributive property.

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) + (- 4 (- x^{2}) - 4 \cdot 25) ((x^{2} + 25) x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.2

Apply the distributive property.

$f'' (x) = \frac{{(x^{2} + 25)}^{2} (- 2 x) + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3

Simplify the numerator.

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Step 2.2.5.3.1

Simplify each term.

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Step 2.2.5.3.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 2 {(x^{2} + 25)}^{2} x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.2

Rewrite ${(x^{2} + 25)}^{2}$ as $(x^{2} + 25) (x^{2} + 25)$ .

$f'' (x) = \frac{- 2 ((x^{2} + 25) (x^{2} + 25)) x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.3

Expand $(x^{2} + 25) (x^{2} + 25)$ using the FOIL Method.

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Step 2.2.5.3.1.3.1

Apply the distributive property.

$f'' (x) = \frac{- 2 (x^{2} (x^{2} + 25) + 25 (x^{2} + 25)) x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.3.2

Apply the distributive property.

$f'' (x) = \frac{- 2 (x^{2} x^{2} + x^{2} \cdot 25 + 25 (x^{2} + 25)) x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.3.3

Apply the distributive property.

$f'' (x) = \frac{- 2 (x^{2} x^{2} + x^{2} \cdot 25 + 25 x^{2} + 25 \cdot 25) x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.4

Simplify and combine like terms.

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Step 2.2.5.3.1.4.1

Simplify each term.

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Step 2.2.5.3.1.4.1.1

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

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Step 2.2.5.3.1.4.1.1.1

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

$f'' (x) = \frac{- 2 (x^{2 + 2} + x^{2} \cdot 25 + 25 x^{2} + 25 \cdot 25) x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.4.1.1.2

Add $2$ and $2$ .

$f'' (x) = \frac{- 2 (x^{4} + x^{2} \cdot 25 + 25 x^{2} + 25 \cdot 25) x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.4.1.2

Move $25$ to the left of $x^{2}$ .

$f'' (x) = \frac{- 2 (x^{4} + 25 \cdot x^{2} + 25 x^{2} + 25 \cdot 25) x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.4.1.3

Multiply $25$ by $25$ .

$f'' (x) = \frac{- 2 (x^{4} + 25 x^{2} + 25 x^{2} + 625) x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.4.2

Add $25 x^{2}$ and $25 x^{2}$ .

$f'' (x) = \frac{- 2 (x^{4} + 50 x^{2} + 625) x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.5

Apply the distributive property.

$f'' (x) = \frac{(- 2 x^{4} - 2 (50 x^{2}) - 2 \cdot 625) x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.6

Simplify.

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Step 2.2.5.3.1.6.1

Multiply $50$ by $- 2$ .

$f'' (x) = \frac{(- 2 x^{4} - 100 x^{2} - 2 \cdot 625) x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.6.2

Multiply $- 2$ by $625$ .

$f'' (x) = \frac{(- 2 x^{4} - 100 x^{2} - 1250) x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.7

Apply the distributive property.

$f'' (x) = \frac{- 2 x^{4} x - 100 x^{2} x - 1250 x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.8

Simplify.

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Step 2.2.5.3.1.8.1

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 2.2.5.3.1.8.1.1

Move $x$ .

$f'' (x) = \frac{- 2 (x \cdot x^{4}) - 100 x^{2} x - 1250 x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.8.1.2

Multiply $x$ by $x^{4}$ .

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Step 2.2.5.3.1.8.1.2.1

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

$f'' (x) = \frac{- 2 (x \cdot x^{4}) - 100 x^{2} x - 1250 x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.8.1.2.2

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

$f'' (x) = \frac{- 2 x^{1 + 4} - 100 x^{2} x - 1250 x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.8.1.3

Add $1$ and $4$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{2} x - 1250 x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.8.2

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

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Step 2.2.5.3.1.8.2.1

Move $x$ .

$f'' (x) = \frac{- 2 x^{5} - 100 (x \cdot x^{2}) - 1250 x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.8.2.2

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

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Step 2.2.5.3.1.8.2.2.1

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

$f'' (x) = \frac{- 2 x^{5} - 100 (x \cdot x^{2}) - 1250 x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.8.2.2.2

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

$f'' (x) = \frac{- 2 x^{5} - 100 x^{1 + 2} - 1250 x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.8.2.3

Add $1$ and $2$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + (- 4 (- x^{2}) - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.9

Simplify each term.

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Step 2.2.5.3.1.9.1

Multiply $- 1$ by $- 4$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + (4 x^{2} - 4 \cdot 25) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.9.2

Multiply $- 4$ by $25$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + (4 x^{2} - 100) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.10

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

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Step 2.2.5.3.1.10.1

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

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Step 2.2.5.3.1.10.1.1

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

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + (4 x^{2} - 100) (x^{2} x + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.10.1.2

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

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + (4 x^{2} - 100) (x^{2 + 1} + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.10.2

Add $2$ and $1$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + (4 x^{2} - 100) (x^{3} + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.11

Expand $(4 x^{2} - 100) (x^{3} + 25 x)$ using the FOIL Method.

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Step 2.2.5.3.1.11.1

Apply the distributive property.

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{2} (x^{3} + 25 x) - 100 (x^{3} + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.11.2

Apply the distributive property.

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{2} x^{3} + 4 x^{2} (25 x) - 100 (x^{3} + 25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.11.3

Apply the distributive property.

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{2} x^{3} + 4 x^{2} (25 x) - 100 x^{3} - 100 (25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.12

Simplify and combine like terms.

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Step 2.2.5.3.1.12.1

Simplify each term.

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Step 2.2.5.3.1.12.1.1

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

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Step 2.2.5.3.1.12.1.1.1

Move $x^{3}$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 (x^{3} x^{2}) + 4 x^{2} (25 x) - 100 x^{3} - 100 (25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.12.1.1.2

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

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{3 + 2} + 4 x^{2} (25 x) - 100 x^{3} - 100 (25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.12.1.1.3

Add $3$ and $2$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{5} + 4 x^{2} (25 x) - 100 x^{3} - 100 (25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.12.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{5} + 4 \cdot (25 x^{2} x) - 100 x^{3} - 100 (25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.12.1.3

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

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Step 2.2.5.3.1.12.1.3.1

Move $x$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{5} + 4 \cdot (25 (x \cdot x^{2})) - 100 x^{3} - 100 (25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.12.1.3.2

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

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Step 2.2.5.3.1.12.1.3.2.1

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

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{5} + 4 \cdot (25 (x \cdot x^{2})) - 100 x^{3} - 100 (25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.12.1.3.2.2

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

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{5} + 4 \cdot (25 x^{1 + 2}) - 100 x^{3} - 100 (25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.12.1.3.3

Add $1$ and $2$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{5} + 4 \cdot (25 x^{3}) - 100 x^{3} - 100 (25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.12.1.4

Multiply $4$ by $25$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{5} + 100 x^{3} - 100 x^{3} - 100 (25 x)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.12.1.5

Multiply $25$ by $- 100$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{5} + 100 x^{3} - 100 x^{3} - 2500 x}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.12.2

Subtract $100 x^{3}$ from $100 x^{3}$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{5} + 0 - 2500 x}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.1.12.3

Add $4 x^{5}$ and $0$ .

$f'' (x) = \frac{- 2 x^{5} - 100 x^{3} - 1250 x + 4 x^{5} - 2500 x}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.2

Add $- 2 x^{5}$ and $4 x^{5}$ .

$f'' (x) = \frac{2 x^{5} - 100 x^{3} - 1250 x - 2500 x}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.3.3

Subtract $2500 x$ from $- 1250 x$ .

$f'' (x) = \frac{2 x^{5} - 100 x^{3} - 3750 x}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.4

Simplify the numerator.

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Step 2.2.5.4.1

Factor $2 x$ out of $2 x^{5} - 100 x^{3} - 3750 x$ .

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Step 2.2.5.4.1.1

Factor $2 x$ out of $2 x^{5}$ .

$f'' (x) = \frac{2 x (x^{4}) - 100 x^{3} - 3750 x}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.4.1.2

Factor $2 x$ out of $- 100 x^{3}$ .

$f'' (x) = \frac{2 x (x^{4}) + 2 x (- 50 x^{2}) - 3750 x}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.4.1.3

Factor $2 x$ out of $- 3750 x$ .

$f'' (x) = \frac{2 x (x^{4}) + 2 x (- 50 x^{2}) + 2 x (- 1875)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.4.1.4

Factor $2 x$ out of $2 x (x^{4}) + 2 x (- 50 x^{2})$ .

$f'' (x) = \frac{2 x (x^{4} - 50 x^{2}) + 2 x (- 1875)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.4.1.5

Factor $2 x$ out of $2 x (x^{4} - 50 x^{2}) + 2 x (- 1875)$ .

$f'' (x) = \frac{2 x (x^{4} - 50 x^{2} - 1875)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.4.2

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

$f'' (x) = \frac{2 x ({(x^{2})}^{2} - 50 x^{2} - 1875)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.4.3

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

$f'' (x) = \frac{2 x (u^{2} - 50 u - 1875)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.4.4

Factor $u^{2} - 50 u - 1875$ using the AC method.

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Step 2.2.5.4.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 1875$ and whose sum is $- 50$ .

$- 75, 25$

Step 2.2.5.4.4.2

Write the factored form using these integers.

$f'' (x) = \frac{2 x ((u - 75) (u + 25))}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.4.5

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

$f'' (x) = \frac{2 x (x^{2} - 75) (x^{2} + 25)}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.5

Cancel the common factor of $x^{2} + 25$ and ${(x^{2} + 25)}^{4}$ .

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Step 2.2.5.5.1

Factor $x^{2} + 25$ out of $2 x (x^{2} - 75) (x^{2} + 25)$ .

$f'' (x) = \frac{(x^{2} + 25) (2 x (x^{2} - 75))}{{(x^{2} + 25)}^{4}}$

Step 2.2.5.5.2

Cancel the common factors.

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Step 2.2.5.5.2.1

Factor $x^{2} + 25$ out of ${(x^{2} + 25)}^{4}$ .

$f'' (x) = \frac{(x^{2} + 25) (2 x (x^{2} - 75))}{(x^{2} + 25) {(x^{2} + 25)}^{3}}$

Step 2.2.5.5.2.2

Cancel the common factor.

$f'' (x) = \frac{(x^{2} + 25) (2 x (x^{2} - 75))}{(x^{2} + 25) {(x^{2} + 25)}^{3}}$

Step 2.2.5.5.2.3

Rewrite the expression.

$f'' (x) = \frac{2 x (x^{2} - 75)}{{(x^{2} + 25)}^{3}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2 x (x^{2} - 75)}{{(x^{2} + 25)}^{3}}$ .

$\frac{2 x (x^{2} - 75)}{{(x^{2} + 25)}^{3}}$

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{2 x (x^{2} - 75)}{{(x^{2} + 25)}^{3}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$\frac{2 x (x^{2} - 75)}{{(x^{2} + 25)}^{3}} = 0$

Step 3.2

Set the numerator equal to zero.

$2 x (x^{2} - 75) = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

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

$x = 0$

$x^{2} - 75 = 0$

Step 3.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 3.3.3

Set $x^{2} - 75$ equal to $0$ and solve for $x$ .

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Step 3.3.3.1

Set $x^{2} - 75$ equal to $0$ .

$x^{2} - 75 = 0$

Step 3.3.3.2

Solve $x^{2} - 75 = 0$ for $x$ .

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Step 3.3.3.2.1

Add $75$ to both sides of the equation.

$x^{2} = 75$

Step 3.3.3.2.2

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

$x = \pm \sqrt{75}$

Step 3.3.3.2.3

Simplify $\pm \sqrt{75}$ .

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Step 3.3.3.2.3.1

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 3.3.3.2.3.1.1

Factor $25$ out of $75$ .

$x = \pm \sqrt{25 (3)}$

Step 3.3.3.2.3.1.2

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2} \cdot 3}$

Step 3.3.3.2.3.2

Pull terms out from under the radical.

$x = \pm 5 \sqrt{3}$

Step 3.3.3.2.4

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

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Step 3.3.3.2.4.1

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

$x = 5 \sqrt{3}$

Step 3.3.3.2.4.2

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

$x = - 5 \sqrt{3}$

Step 3.3.3.2.4.3

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

$x = 5 \sqrt{3}, - 5 \sqrt{3}$

Step 3.3.4

The final solution is all the values that make $2 x (x^{2} - 75) = 0$ true.

$x = 0, 5 \sqrt{3}, - 5 \sqrt{3}$

Step 4

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

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Step 4.1

Substitute $0$ in $f (x) = \frac{x}{x^{2} + 25}$ to find the value of $y$ .

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Step 4.1.1

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

$f (0) = \frac{0}{{(0)}^{2} + 25}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify the denominator.

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Step 4.1.2.1.1

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

$f (0) = \frac{0}{0 + 25}$

Step 4.1.2.1.2

Add $0$ and $25$ .

$f (0) = \frac{0}{25}$

Step 4.1.2.2

Divide $0$ by $25$ .

$f (0) = 0$

Step 4.1.2.3

The final answer is $0$ .

$0$

Step 4.2

The point found by substituting $0$ in $f (x) = \frac{x}{x^{2} + 25}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 4.3

Substitute $5 \sqrt{3}$ in $f (x) = \frac{x}{x^{2} + 25}$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $5 \sqrt{3}$ in the expression.

$f (5 \sqrt{3}) = \frac{5 \sqrt{3}}{{(5 \sqrt{3})}^{2} + 25}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify the denominator.

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Step 4.3.2.1.1

Apply the product rule to $5 \sqrt{3}$ .

$f (5 \sqrt{3}) = \frac{5 \sqrt{3}}{5^{2} {\sqrt{3}}^{2} + 25}$

Step 4.3.2.1.2

Raise $5$ to the power of $2$ .

$f (5 \sqrt{3}) = \frac{5 \sqrt{3}}{25 {\sqrt{3}}^{2} + 25}$

Step 4.3.2.1.3

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

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Step 4.3.2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (5 \sqrt{3}) = \frac{5 \sqrt{3}}{25 {(3^{\frac{1}{2}})}^{2} + 25}$

Step 4.3.2.1.3.2

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

$f (5 \sqrt{3}) = \frac{5 \sqrt{3}}{25 \cdot 3^{\frac{1}{2} \cdot 2} + 25}$

Step 4.3.2.1.3.3

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

$f (5 \sqrt{3}) = \frac{5 \sqrt{3}}{25 \cdot 3^{\frac{2}{2}} + 25}$

Step 4.3.2.1.3.4

Cancel the common factor of $2$ .

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Step 4.3.2.1.3.4.1

Cancel the common factor.

$f (5 \sqrt{3}) = \frac{5 \sqrt{3}}{25 \cdot 3^{\frac{2}{2}} + 25}$

Step 4.3.2.1.3.4.2

Rewrite the expression.

$f (5 \sqrt{3}) = \frac{5 \sqrt{3}}{25 \cdot 3 + 25}$

Step 4.3.2.1.3.5

Evaluate the exponent.

$f (5 \sqrt{3}) = \frac{5 \sqrt{3}}{25 \cdot 3 + 25}$

Step 4.3.2.1.4

Multiply $25$ by $3$ .

$f (5 \sqrt{3}) = \frac{5 \sqrt{3}}{75 + 25}$

Step 4.3.2.1.5

Add $75$ and $25$ .

$f (5 \sqrt{3}) = \frac{5 \sqrt{3}}{100}$

Step 4.3.2.2

Cancel the common factor of $5$ and $100$ .

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Step 4.3.2.2.1

Factor $5$ out of $5 \sqrt{3}$ .

$f (5 \sqrt{3}) = \frac{5 (\sqrt{3})}{100}$

Step 4.3.2.2.2

Cancel the common factors.

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Step 4.3.2.2.2.1

Factor $5$ out of $100$ .

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

Step 4.3.2.2.2.2

Cancel the common factor.

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

Step 4.3.2.2.2.3

Rewrite the expression.

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

Step 4.3.2.3

The final answer is $\frac{\sqrt{3}}{20}$ .

$\frac{\sqrt{3}}{20}$

Step 4.4

The point found by substituting $5 \sqrt{3}$ in $f (x) = \frac{x}{x^{2} + 25}$ is $(5 \sqrt{3}, \frac{\sqrt{3}}{20})$ . This point can be an inflection point.

$(5 \sqrt{3}, \frac{\sqrt{3}}{20})$

Step 4.5

Substitute $- 5 \sqrt{3}$ in $f (x) = \frac{x}{x^{2} + 25}$ to find the value of $y$ .

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Step 4.5.1

Replace the variable $x$ with $- 5 \sqrt{3}$ in the expression.

$f (- 5 \sqrt{3}) = \frac{- 5 \sqrt{3}}{{(- 5 \sqrt{3})}^{2} + 25}$

Step 4.5.2

Simplify the result.

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Step 4.5.2.1

Simplify the denominator.

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Step 4.5.2.1.1

Apply the product rule to $- 5 \sqrt{3}$ .

$f (- 5 \sqrt{3}) = \frac{- 5 \sqrt{3}}{{(- 5)}^{2} {\sqrt{3}}^{2} + 25}$

Step 4.5.2.1.2

Raise $- 5$ to the power of $2$ .

$f (- 5 \sqrt{3}) = \frac{- 5 \sqrt{3}}{25 {\sqrt{3}}^{2} + 25}$

Step 4.5.2.1.3

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

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Step 4.5.2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (- 5 \sqrt{3}) = \frac{- 5 \sqrt{3}}{25 {(3^{\frac{1}{2}})}^{2} + 25}$

Step 4.5.2.1.3.2

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

$f (- 5 \sqrt{3}) = \frac{- 5 \sqrt{3}}{25 \cdot 3^{\frac{1}{2} \cdot 2} + 25}$

Step 4.5.2.1.3.3

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

$f (- 5 \sqrt{3}) = \frac{- 5 \sqrt{3}}{25 \cdot 3^{\frac{2}{2}} + 25}$

Step 4.5.2.1.3.4

Cancel the common factor of $2$ .

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Step 4.5.2.1.3.4.1

Cancel the common factor.

$f (- 5 \sqrt{3}) = \frac{- 5 \sqrt{3}}{25 \cdot 3^{\frac{2}{2}} + 25}$

Step 4.5.2.1.3.4.2

Rewrite the expression.

$f (- 5 \sqrt{3}) = \frac{- 5 \sqrt{3}}{25 \cdot 3 + 25}$

Step 4.5.2.1.3.5

Evaluate the exponent.

$f (- 5 \sqrt{3}) = \frac{- 5 \sqrt{3}}{25 \cdot 3 + 25}$

Step 4.5.2.1.4

Multiply $25$ by $3$ .

$f (- 5 \sqrt{3}) = \frac{- 5 \sqrt{3}}{75 + 25}$

Step 4.5.2.1.5

Add $75$ and $25$ .

$f (- 5 \sqrt{3}) = \frac{- 5 \sqrt{3}}{100}$

Step 4.5.2.2

Reduce the expression by cancelling the common factors.

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Step 4.5.2.2.1

Cancel the common factor of $- 5$ and $100$ .

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Step 4.5.2.2.1.1

Factor $5$ out of $- 5 \sqrt{3}$ .

$f (- 5 \sqrt{3}) = \frac{5 (- \sqrt{3})}{100}$

Step 4.5.2.2.1.2

Cancel the common factors.

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Step 4.5.2.2.1.2.1

Factor $5$ out of $100$ .

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

Step 4.5.2.2.1.2.2

Cancel the common factor.

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

Step 4.5.2.2.1.2.3

Rewrite the expression.

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

Step 4.5.2.2.2

Move the negative in front of the fraction.

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

Step 4.5.2.3

The final answer is $- \frac{\sqrt{3}}{20}$ .

$- \frac{\sqrt{3}}{20}$

Step 4.6

The point found by substituting $- 5 \sqrt{3}$ in $f (x) = \frac{x}{x^{2} + 25}$ is $(- 5 \sqrt{3}, - \frac{\sqrt{3}}{20})$ . This point can be an inflection point.

$(- 5 \sqrt{3}, - \frac{\sqrt{3}}{20})$

Step 4.7

Determine the points that could be inflection points.

$(0, 0), (5 \sqrt{3}, \frac{\sqrt{3}}{20}), (- 5 \sqrt{3}, - \frac{\sqrt{3}}{20})$

Step 5

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

$(- \infty, - 5 \sqrt{3}) \cup (- 5 \sqrt{3}, 0) \cup (0, 5 \sqrt{3}) \cup (5 \sqrt{3}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 5 \sqrt{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 8.76025403$ in the expression.

$f'' (- 8.76025403) = \frac{2 (- 8.76025403) ({(- 8.76025403)}^{2} - 75)}{{({(- 8.76025403)}^{2} + 25)}^{3}}$

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 $2$ by $- 8.76025403$ .

$f'' (- 8.76025403) = \frac{- 17.52050807 ({(- 8.76025403)}^{2} - 75)}{{({(- 8.76025403)}^{2} + 25)}^{3}}$

Step 6.2.1.2

Multiply $- 17.52050807$ by ${(- 8.76025403)}^{2} - 75$ .

$f'' (- 8.76025403) = \frac{- 30.52161524}{{({(- 8.76025403)}^{2} + 25)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Raise $- 8.76025403$ to the power of $2$ .

$f'' (- 8.76025403) = \frac{- 30.52161524}{{(76.7420508 + 25)}^{3}}$

Step 6.2.2.2

Add $76.7420508$ and $25$ .

$f'' (- 8.76025403) = \frac{- 30.52161524}{{101.7420508}^{3}}$

Step 6.2.2.3

Raise $101.7420508$ to the power of $3$ .

$f'' (- 8.76025403) = \frac{- 30.52161524}{1053177.23320495}$

Step 6.2.3

Divide $- 30.52161524$ by $1053177.23320495$ .

$f'' (- 8.76025403) = - 0.00002898$

Step 6.2.4

The final answer is $- 0.00002898$ .

$- 0.00002898$

Step 6.3

At $- 8.76025403$ , the second derivative is $- 2.89805118 E - 5$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 5 \sqrt{3})$

Decreasing on $(- \infty, - 5 \sqrt{3})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 5 \sqrt{3}, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 4.33012701$ in the expression.

$f'' (- 4.33012701) = \frac{2 (- 4.33012701) ({(- 4.33012701)}^{2} - 75)}{{({(- 4.33012701)}^{2} + 25)}^{3}}$

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 $2$ by $- 4.33012701$ .

$f'' (- 4.33012701) = \frac{- 8.66025403 ({(- 4.33012701)}^{2} - 75)}{{({(- 4.33012701)}^{2} + 25)}^{3}}$

Step 7.2.1.2

Multiply $- 8.66025403$ by ${(- 4.33012701)}^{2} - 75$ .

$f'' (- 4.33012701) = \frac{487.13928962}{{({(- 4.33012701)}^{2} + 25)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Raise $- 4.33012701$ to the power of $2$ .

$f'' (- 4.33012701) = \frac{487.13928962}{{(18.75 + 25)}^{3}}$

Step 7.2.2.2

Add $18.75$ and $25$ .

$f'' (- 4.33012701) = \frac{487.13928962}{{43.75}^{3}}$

Step 7.2.2.3

Raise $43.75$ to the power of $3$ .

$f'' (- 4.33012701) = \frac{487.13928962}{83740.234375}$

Step 7.2.3

Divide $487.13928962$ by $83740.234375$ .

$f'' (- 4.33012701) = 0.00581726$

Step 7.2.4

The final answer is $0.00581726$ .

$0.00581726$

Step 7.3

At $- 4.33012701$ , the second derivative is $0.00581726$ . Since this is positive, the second derivative is increasing on the interval $(- 5 \sqrt{3}, 0)$ .

Increasing on $(- 5 \sqrt{3}, 0)$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(0, 5 \sqrt{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $4.33012701$ in the expression.

$f'' (4.33012701) = \frac{2 (4.33012701) ({(4.33012701)}^{2} - 75)}{{({(4.33012701)}^{2} + 25)}^{3}}$

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 $2$ by $4.33012701$ .

$f'' (4.33012701) = \frac{8.66025403 ({4.33012701}^{2} - 75)}{{({4.33012701}^{2} + 25)}^{3}}$

Step 8.2.1.2

Multiply $8.66025403$ by ${4.33012701}^{2} - 75$ .

$f'' (4.33012701) = \frac{- 487.13928962}{{({4.33012701}^{2} + 25)}^{3}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Raise $4.33012701$ to the power of $2$ .

$f'' (4.33012701) = \frac{- 487.13928962}{{(18.75 + 25)}^{3}}$

Step 8.2.2.2

Add $18.75$ and $25$ .

$f'' (4.33012701) = \frac{- 487.13928962}{{43.75}^{3}}$

Step 8.2.2.3

Raise $43.75$ to the power of $3$ .

$f'' (4.33012701) = \frac{- 487.13928962}{83740.234375}$

Step 8.2.3

Divide $- 487.13928962$ by $83740.234375$ .

$f'' (4.33012701) = - 0.00581726$

Step 8.2.4

The final answer is $- 0.00581726$ .

$- 0.00581726$

Step 8.3

At $4.33012701$ , the second derivative is $- 0.00581726$ . Since this is negative, the second derivative is decreasing on the interval $(0, 5 \sqrt{3})$

Decreasing on $(0, 5 \sqrt{3})$ since $f'' (x) < 0$

Step 9

Substitute a value from the interval $(5 \sqrt{3}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $8.76025403$ in the expression.

$f'' (8.76025403) = \frac{2 (8.76025403) ({(8.76025403)}^{2} - 75)}{{({(8.76025403)}^{2} + 25)}^{3}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify the numerator.

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Step 9.2.1.1

Multiply $2$ by $8.76025403$ .

$f'' (8.76025403) = \frac{17.52050807 ({8.76025403}^{2} - 75)}{{({8.76025403}^{2} + 25)}^{3}}$

Step 9.2.1.2

Multiply $17.52050807$ by ${8.76025403}^{2} - 75$ .

$f'' (8.76025403) = \frac{30.52161524}{{({8.76025403}^{2} + 25)}^{3}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Raise $8.76025403$ to the power of $2$ .

$f'' (8.76025403) = \frac{30.52161524}{{(76.7420508 + 25)}^{3}}$

Step 9.2.2.2

Add $76.7420508$ and $25$ .

$f'' (8.76025403) = \frac{30.52161524}{{101.7420508}^{3}}$

Step 9.2.2.3

Raise $101.7420508$ to the power of $3$ .

$f'' (8.76025403) = \frac{30.52161524}{1053177.23320495}$

Step 9.2.3

Divide $30.52161524$ by $1053177.23320495$ .

$f'' (8.76025403) = 0.00002898$

Step 9.2.4

The final answer is $0.00002898$ .

$0.00002898$

Step 9.3

At $8.76025403$ , the second derivative is $2.89805118 E - 5$ . Since this is positive, the second derivative is increasing on the interval $(5 \sqrt{3}, \infty)$ .

Increasing on $(5 \sqrt{3}, \infty)$ since $f'' (x) > 0$

Step 10

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 $(- 5 \sqrt{3}, - \frac{\sqrt{3}}{20}), (0, 0), (5 \sqrt{3}, \frac{\sqrt{3}}{20})$ .

$(- 5 \sqrt{3}, - \frac{\sqrt{3}}{20}), (0, 0), (5 \sqrt{3}, \frac{\sqrt{3}}{20})$

Step 11