Calculus Examples

$y = x + \frac{250}{x^{2} + 25}$

Step 1

Write $y = x + \frac{250}{x^{2} + 25}$ as a function.

$f (x) = x + \frac{250}{x^{2} + 25}$

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x + \frac{250}{x^{2} + 25}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{250}{x^{2} + 25}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{250}{x^{2} + 25}]$

Step 2.1.2

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

$1 + \frac{d}{d x} [\frac{250}{x^{2} + 25}]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{250}{x^{2} + 25}]$ .

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Step 2.2.1

Since $250$ is constant with respect to $x$ , the derivative of $\frac{250}{x^{2} + 25}$ with respect to $x$ is $250 \frac{d}{d x} [\frac{1}{x^{2} + 25}]$ .

$1 + 250 \frac{d}{d x} [\frac{1}{x^{2} + 25}]$

Step 2.2.2

Rewrite $\frac{1}{x^{2} + 25}$ as ${(x^{2} + 25)}^{- 1}$ .

$1 + 250 \frac{d}{d x} [{(x^{2} + 25)}^{- 1}]$

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 $f (x) = x^{- 1}$ and $g (x) = x^{2} + 25$ .

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Step 2.2.3.1

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

$1 + 250 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} + 25])$

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 $n = - 1$ .

$1 + 250 (- u^{- 2} \frac{d}{d x} [x^{2} + 25])$

Step 2.2.3.3

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

$1 + 250 (- {(x^{2} + 25)}^{- 2} \frac{d}{d x} [x^{2} + 25])$

Step 2.2.4

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

$1 + 250 (- {(x^{2} + 25)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [25]))$

Step 2.2.5

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

$1 + 250 (- {(x^{2} + 25)}^{- 2} (2 x + \frac{d}{d x} [25]))$

Step 2.2.6

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

$1 + 250 (- {(x^{2} + 25)}^{- 2} (2 x + 0))$

Step 2.2.7

Add $2 x$ and $0$ .

$1 + 250 (- {(x^{2} + 25)}^{- 2} (2 x))$

Step 2.2.8

Multiply $2$ by $- 1$ .

$1 + 250 (- 2 {(x^{2} + 25)}^{- 2} x)$

Step 2.2.9

Multiply $- 2$ by $250$ .

$1 - 500 {(x^{2} + 25)}^{- 2} x$

Step 2.3

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

$1 - 500 \frac{1}{{(x^{2} + 25)}^{2}} x$

Step 2.4

Simplify.

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Step 2.4.1

Combine terms.

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Step 2.4.1.1

Combine $- 500$ and $\frac{1}{{(x^{2} + 25)}^{2}}$ .

$1 + \frac{- 500}{{(x^{2} + 25)}^{2}} x$

Step 2.4.1.2

Move the negative in front of the fraction.

$1 - \frac{500}{{(x^{2} + 25)}^{2}} x$

Step 2.4.1.3

Combine $x$ and $\frac{500}{{(x^{2} + 25)}^{2}}$ .

$1 - \frac{x \cdot 500}{{(x^{2} + 25)}^{2}}$

Step 2.4.1.4

Move $500$ to the left of $x$ .

$1 - \frac{500 x}{{(x^{2} + 25)}^{2}}$

Step 2.4.2

Reorder terms.

$- \frac{500 x}{{(x^{2} + 25)}^{2}} + 1$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $- \frac{500 x}{{(x^{2} + 25)}^{2}} + 1$ with respect to $x$ is $\frac{d}{d x} [- \frac{500 x}{{(x^{2} + 25)}^{2}}] + \frac{d}{d x} [1]$ .

$f'' (x) = \frac{d}{d x} (- \frac{500 x}{{(x^{2} + 25)}^{2}}) + \frac{d}{d x} (1)$

Step 3.2

Evaluate $\frac{d}{d x} [- \frac{500 x}{{(x^{2} + 25)}^{2}}]$ .

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Step 3.2.1

Since $- 500$ is constant with respect to $x$ , the derivative of $- \frac{500 x}{{(x^{2} + 25)}^{2}}$ with respect to $x$ is $- 500 \frac{d}{d x} [\frac{x}{{(x^{2} + 25)}^{2}}]$ .

$f'' (x) = - 500 \frac{d}{d x} \frac{x}{{(x^{2} + 25)}^{2}} + \frac{d}{d x} (1)$

Step 3.2.2

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 25)}^{2}}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.3

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} \cdot 1 - x \frac{d}{d x} {(x^{2} + 25)}^{2}}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.4

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

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Step 3.2.4.1

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} \cdot 1 - x (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} + 25))}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.4.2

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} \cdot 1 - x (2 u \frac{d}{d x} (x^{2} + 25))}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.4.3

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} \cdot 1 - x (2 (x^{2} + 25) \frac{d}{d x} (x^{2} + 25))}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.5

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} \cdot 1 - x (2 (x^{2} + 25) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (25)))}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.6

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} \cdot 1 - x (2 (x^{2} + 25) (2 x + \frac{d}{d x} (25)))}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.7

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} \cdot 1 - x (2 (x^{2} + 25) (2 x + 0))}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.8

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} - x (2 (x^{2} + 25) (2 x + 0))}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.9

Add $2 x$ and $0$ .

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} - x (2 (x^{2} + 25) (2 x))}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.10

Multiply $2$ by $2$ .

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} - x (4 (x^{2} + 25) x)}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.11

Multiply $4$ by $- 1$ .

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} - 4 x ((x^{2} + 25) x)}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.12

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} - 4 (x \cdot x) (x^{2} + 25)}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.13

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} - 4 (x \cdot x) (x^{2} + 25)}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.14

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} - 4 x^{1 + 1} (x^{2} + 25)}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.15

Add $1$ and $1$ .

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} - 4 x^{2} (x^{2} + 25)}{{({(x^{2} + 25)}^{2})}^{2}} + \frac{d}{d x} (1)$

Step 3.2.16

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

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Step 3.2.16.1

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

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} - 4 x^{2} (x^{2} + 25)}{{(x^{2} + 25)}^{2 \cdot 2}} + \frac{d}{d x} (1)$

Step 3.2.16.2

Multiply $2$ by $2$ .

$f'' (x) = - 500 \frac{{(x^{2} + 25)}^{2} - 4 x^{2} (x^{2} + 25)}{{(x^{2} + 25)}^{4}} + \frac{d}{d x} (1)$

Step 3.2.17

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

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Step 3.2.17.1

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

$f'' (x) = - 500 \frac{(x^{2} + 25) (x^{2} + 25) - 4 x^{2} (x^{2} + 25)}{{(x^{2} + 25)}^{4}} + \frac{d}{d x} (1)$

Step 3.2.17.2

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

$f'' (x) = - 500 \frac{(x^{2} + 25) (x^{2} + 25) + (x^{2} + 25) (- 4 x^{2})}{{(x^{2} + 25)}^{4}} + \frac{d}{d x} (1)$

Step 3.2.17.3

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

$f'' (x) = - 500 \frac{(x^{2} + 25) (x^{2} + 25 - 4 x^{2})}{{(x^{2} + 25)}^{4}} + \frac{d}{d x} (1)$

Step 3.2.18

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

$f'' (x) = - 500 \frac{(x^{2} + 25) (- 3 x^{2} + 25)}{{(x^{2} + 25)}^{4}} + \frac{d}{d x} (1)$

Step 3.2.19

Cancel the common factors.

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Step 3.2.19.1

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

$f'' (x) = - 500 \frac{(x^{2} + 25) (- 3 x^{2} + 25)}{(x^{2} + 25) {(x^{2} + 25)}^{3}} + \frac{d}{d x} (1)$

Step 3.2.19.2

Cancel the common factor.

$f'' (x) = - 500 \frac{(x^{2} + 25) (- 3 x^{2} + 25)}{(x^{2} + 25) {(x^{2} + 25)}^{3}} + \frac{d}{d x} (1)$

Step 3.2.19.3

Rewrite the expression.

$f'' (x) = - 500 \frac{- 3 x^{2} + 25}{{(x^{2} + 25)}^{3}} + \frac{d}{d x} (1)$

Step 3.2.20

Combine $- 500$ and $\frac{- 3 x^{2} + 25}{{(x^{2} + 25)}^{3}}$ .

$f'' (x) = \frac{- 500 (- 3 x^{2} + 25)}{{(x^{2} + 25)}^{3}} + \frac{d}{d x} (1)$

Step 3.2.21

Move the negative in front of the fraction.

$f'' (x) = - \frac{500 (- 3 x^{2} + 25)}{{(x^{2} + 25)}^{3}} + \frac{d}{d x} (1)$

Step 3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{500 (- 3 x^{2} + 25)}{{(x^{2} + 25)}^{3}} + 0$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$f'' (x) = - \frac{500 (- 3 x^{2}) + 500 \cdot 25}{{(x^{2} + 25)}^{3}} + 0$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Multiply $- 3$ by $500$ .

$f'' (x) = - \frac{- 1500 x^{2} + 500 \cdot 25}{{(x^{2} + 25)}^{3}} + 0$

Step 3.4.2.2

Multiply $500$ by $25$ .

$f'' (x) = - \frac{- 1500 x^{2} + 12500}{{(x^{2} + 25)}^{3}} + 0$

Step 3.4.2.3

Add $- \frac{- 1500 x^{2} + 12500}{{(x^{2} + 25)}^{3}}$ and $0$ .

$f'' (x) = - \frac{- 1500 x^{2} + 12500}{{(x^{2} + 25)}^{3}}$

Step 3.4.3

Factor $500$ out of $- 1500 x^{2} + 12500$ .

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Step 3.4.3.1

Factor $500$ out of $- 1500 x^{2}$ .

$f'' (x) = - \frac{500 (- 3 x^{2}) + 12500}{{(x^{2} + 25)}^{3}}$

Step 3.4.3.2

Factor $500$ out of $12500$ .

$f'' (x) = - \frac{500 (- 3 x^{2}) + 500 (25)}{{(x^{2} + 25)}^{3}}$

Step 3.4.3.3

Factor $500$ out of $500 (- 3 x^{2}) + 500 (25)$ .

$f'' (x) = - \frac{500 (- 3 x^{2} + 25)}{{(x^{2} + 25)}^{3}}$

Step 3.4.4

Factor $- 1$ out of $- 3 x^{2}$ .

$f'' (x) = - \frac{500 (- (3 x^{2}) + 25)}{{(x^{2} + 25)}^{3}}$

Step 3.4.5

Rewrite $25$ as $- 1 (- 25)$ .

$f'' (x) = - \frac{500 (- (3 x^{2}) - 1 \cdot - 25)}{{(x^{2} + 25)}^{3}}$

Step 3.4.6

Factor $- 1$ out of $- (3 x^{2}) - 1 (- 25)$ .

$f'' (x) = - \frac{500 (- (3 x^{2} - 25))}{{(x^{2} + 25)}^{3}}$

Step 3.4.7

Rewrite $- (3 x^{2} - 25)$ as $- 1 (3 x^{2} - 25)$ .

$f'' (x) = - \frac{500 (- 1 (3 x^{2} - 25))}{{(x^{2} + 25)}^{3}}$

Step 3.4.8

Move the negative in front of the fraction.

$f'' (x) = \frac{500 (3 x^{2} - 25)}{{(x^{2} + 25)}^{3}}$

Step 3.4.9

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{500 (3 x^{2} - 25)}{{(x^{2} + 25)}^{3}})$

Step 3.4.10

Multiply $\frac{500 (3 x^{2} - 25)}{{(x^{2} + 25)}^{3}}$ by $1$ .

$f'' (x) = \frac{500 (3 x^{2} - 25)}{{(x^{2} + 25)}^{3}}$

Step 4

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

$- \frac{500 x}{{(x^{2} + 25)}^{2}} + 1 = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Differentiate.

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Step 5.1.1.1

By the Sum Rule, the derivative of $x + \frac{250}{x^{2} + 25}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{250}{x^{2} + 25}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{250}{x^{2} + 25}]$

Step 5.1.1.2

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

$1 + \frac{d}{d x} [\frac{250}{x^{2} + 25}]$

Step 5.1.2

Evaluate $\frac{d}{d x} [\frac{250}{x^{2} + 25}]$ .

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Step 5.1.2.1

Since $250$ is constant with respect to $x$ , the derivative of $\frac{250}{x^{2} + 25}$ with respect to $x$ is $250 \frac{d}{d x} [\frac{1}{x^{2} + 25}]$ .

$1 + 250 \frac{d}{d x} [\frac{1}{x^{2} + 25}]$

Step 5.1.2.2

Rewrite $\frac{1}{x^{2} + 25}$ as ${(x^{2} + 25)}^{- 1}$ .

$1 + 250 \frac{d}{d x} [{(x^{2} + 25)}^{- 1}]$

Step 5.1.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} + 25$ .

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Step 5.1.2.3.1

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

$1 + 250 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} + 25])$

Step 5.1.2.3.2

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

$1 + 250 (- u^{- 2} \frac{d}{d x} [x^{2} + 25])$

Step 5.1.2.3.3

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

$1 + 250 (- {(x^{2} + 25)}^{- 2} \frac{d}{d x} [x^{2} + 25])$

Step 5.1.2.4

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

$1 + 250 (- {(x^{2} + 25)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [25]))$

Step 5.1.2.5

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

$1 + 250 (- {(x^{2} + 25)}^{- 2} (2 x + \frac{d}{d x} [25]))$

Step 5.1.2.6

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

$1 + 250 (- {(x^{2} + 25)}^{- 2} (2 x + 0))$

Step 5.1.2.7

Add $2 x$ and $0$ .

$1 + 250 (- {(x^{2} + 25)}^{- 2} (2 x))$

Step 5.1.2.8

Multiply $2$ by $- 1$ .

$1 + 250 (- 2 {(x^{2} + 25)}^{- 2} x)$

Step 5.1.2.9

Multiply $- 2$ by $250$ .

$1 - 500 {(x^{2} + 25)}^{- 2} x$

Step 5.1.3

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

$1 - 500 \frac{1}{{(x^{2} + 25)}^{2}} x$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Combine terms.

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Step 5.1.4.1.1

Combine $- 500$ and $\frac{1}{{(x^{2} + 25)}^{2}}$ .

$1 + \frac{- 500}{{(x^{2} + 25)}^{2}} x$

Step 5.1.4.1.2

Move the negative in front of the fraction.

$1 - \frac{500}{{(x^{2} + 25)}^{2}} x$

Step 5.1.4.1.3

Combine $x$ and $\frac{500}{{(x^{2} + 25)}^{2}}$ .

$1 - \frac{x \cdot 500}{{(x^{2} + 25)}^{2}}$

Step 5.1.4.1.4

Move $500$ to the left of $x$ .

$1 - \frac{500 x}{{(x^{2} + 25)}^{2}}$

Step 5.1.4.2

Reorder terms.

$f' (x) = - \frac{500 x}{{(x^{2} + 25)}^{2}} + 1$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{500 x}{{(x^{2} + 25)}^{2}} + 1$ .

$- \frac{500 x}{{(x^{2} + 25)}^{2}} + 1$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{500 x}{{(x^{2} + 25)}^{2}} + 1 = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{500 x}{{(x^{2} + 25)}^{2}} + 1 = 0$

Step 6.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 1.47798871, 5$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 8

Critical points to evaluate.

$x = 1.47798871, 5$

Step 9

Evaluate the second derivative at $x = 1.47798871$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{500 (3 {(1.47798871)}^{2} - 25)}{{({(1.47798871)}^{2} + 25)}^{3}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $1.47798871$ to the power of $2$ .

$\frac{500 (3 \cdot 2.18445063 - 25)}{{({1.47798871}^{2} + 25)}^{3}}$

Step 10.1.2

Multiply $3$ by $2.18445063$ .

$\frac{500 (6.55335191 - 25)}{{({1.47798871}^{2} + 25)}^{3}}$

Step 10.1.3

Subtract $25$ from $6.55335191$ .

$\frac{500 \cdot - 18.44664808}{{({1.47798871}^{2} + 25)}^{3}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Raise $1.47798871$ to the power of $2$ .

$\frac{500 \cdot - 18.44664808}{{(2.18445063 + 25)}^{3}}$

Step 10.2.2

Add $2.18445063$ and $25$ .

$\frac{500 \cdot - 18.44664808}{{27.18445063}^{3}}$

Step 10.2.3

Raise $27.18445063$ to the power of $3$ .

$\frac{500 \cdot - 18.44664808}{20089.1556072}$

Step 10.3

Simplify the expression.

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Step 10.3.1

Multiply $500$ by $- 18.44664808$ .

$\frac{- 9223.32404202}{20089.1556072}$

Step 10.3.2

Divide $- 9223.32404202$ by $20089.1556072$ .

$- 0.45911954$

Step 11

$x = 1.47798871$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 1.47798871$ is a local maximum

Step 12

Find the y-value when $x = 1.47798871$ .

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Step 12.1

Replace the variable $x$ with $1.47798871$ in the expression.

$f (1.47798871) = (1.47798871) + \frac{250}{{(1.47798871)}^{2} + 25}$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Simplify the denominator.

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Step 12.2.1.1.1

Raise $1.47798871$ to the power of $2$ .

$f (1.47798871) = 1.47798871 + \frac{250}{2.18445063 + 25}$

Step 12.2.1.1.2

Add $2.18445063$ and $25$ .

$f (1.47798871) = 1.47798871 + \frac{250}{27.18445063}$

Step 12.2.1.2

Divide $250$ by $27.18445063$ .

$f (1.47798871) = 1.47798871 + 9.19643377$

Step 12.2.2

Add $1.47798871$ and $9.19643377$ .

$f (1.47798871) = 10.67442248$

Step 12.2.3

The final answer is $10.67442248$ .

$y = 10.67442248$

Step 13

Evaluate the second derivative at $x = 5$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{500 (3 {(5)}^{2} - 25)}{{({(5)}^{2} + 25)}^{3}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

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

$\frac{500 (3 \cdot 25 - 25)}{{(5^{2} + 25)}^{3}}$

Step 14.1.2

Multiply $3$ by $25$ .

$\frac{500 (75 - 25)}{{(5^{2} + 25)}^{3}}$

Step 14.1.3

Subtract $25$ from $75$ .

$\frac{500 \cdot 50}{{(5^{2} + 25)}^{3}}$

Step 14.2

Simplify the denominator.

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Step 14.2.1

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

$\frac{500 \cdot 50}{{(25 + 25)}^{3}}$

Step 14.2.2

Add $25$ and $25$ .

$\frac{500 \cdot 50}{50^{3}}$

Step 14.2.3

Raise $50$ to the power of $3$ .

$\frac{500 \cdot 50}{125000}$

Step 14.3

Reduce the expression by cancelling the common factors.

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Step 14.3.1

Multiply $500$ by $50$ .

$\frac{25000}{125000}$

Step 14.3.2

Cancel the common factor of $25000$ and $125000$ .

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Step 14.3.2.1

Factor $25000$ out of $25000$ .

$\frac{25000 (1)}{125000}$

Step 14.3.2.2

Cancel the common factors.

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Step 14.3.2.2.1

Factor $25000$ out of $125000$ .

$\frac{25000 \cdot 1}{25000 \cdot 5}$

Step 14.3.2.2.2

Cancel the common factor.

$\frac{25000 \cdot 1}{25000 \cdot 5}$

Step 14.3.2.2.3

Rewrite the expression.

$\frac{1}{5}$

Step 15

$x = 5$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 5$ is a local minimum

Step 16

Find the y-value when $x = 5$ .

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Step 16.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = (5) + \frac{250}{{(5)}^{2} + 25}$

Step 16.2

Simplify the result.

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Step 16.2.1

Simplify each term.

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Step 16.2.1.1

Simplify the denominator.

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Step 16.2.1.1.1

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

$f (5) = 5 + \frac{250}{25 + 25}$

Step 16.2.1.1.2

Add $25$ and $25$ .

$f (5) = 5 + \frac{250}{50}$

Step 16.2.1.2

Divide $250$ by $50$ .

$f (5) = 5 + 5$

Step 16.2.2

Add $5$ and $5$ .

$f (5) = 10$

Step 16.2.3

The final answer is $10$ .

$y = 10$

Step 17

These are the local extrema for $f (x) = x + \frac{250}{x^{2} + 25}$ .

$(1.47798871, 10.67442248)$ is a local maxima

$(5, 10)$ is a local minima

Step 18