Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Multiply $- 1$ by $2$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

Simplify the expression.

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Step 1.1.3.5.1

Add $2 x$ and $0$ .

$- 2 {(x^{2} - 25)}^{- 2} (2 x)$

Step 1.1.3.5.2

Multiply $2$ by $- 2$ .

$- 4 {(x^{2} - 25)}^{- 2} x$

Step 1.1.4

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

$- 4 \frac{1}{{(x^{2} - 25)}^{2}} x$

Step 1.1.5

Combine terms.

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Step 1.1.5.1

Combine $- 4$ and $\frac{1}{{(x^{2} - 25)}^{2}}$ .

$\frac{- 4}{{(x^{2} - 25)}^{2}} x$

Step 1.1.5.2

Move the negative in front of the fraction.

$- \frac{4}{{(x^{2} - 25)}^{2}} x$

Step 1.1.5.3

Combine $x$ and $\frac{4}{{(x^{2} - 25)}^{2}}$ .

$- \frac{x \cdot 4}{{(x^{2} - 25)}^{2}}$

Step 1.1.5.4

Move $4$ to the left of $x$ .

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{4 x}{{(x^{2} - 25)}^{2}}$ .

$- \frac{4 x}{{(x^{2} - 25)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{4 x}{{(x^{2} - 25)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$4 x = 0$

Step 2.3

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 2.3.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $4$ .

$x = 0$

Step 3

The values which make the derivative equal to $0$ are $0$ .

$0$

Step 4

Find where the derivative is undefined.

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Step 4.1

Set the denominator in $\frac{4 x}{{(x^{2} - 25)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x^{2} - 25)}^{2} = 0$

Step 4.2

Solve for $x$ .

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Step 4.2.1

Factor the left side of the equation.

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Step 4.2.1.1

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

${(x^{2} - 5^{2})}^{2} = 0$

Step 4.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

${((x + 5) (x - 5))}^{2} = 0$

Step 4.2.1.3

Apply the product rule to $(x + 5) (x - 5)$ .

${(x + 5)}^{2} {(x - 5)}^{2} = 0$

Step 4.2.2

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

${(x + 5)}^{2} = 0$

${(x - 5)}^{2} = 0$

Step 4.2.3

Set ${(x + 5)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.2.3.1

Set ${(x + 5)}^{2}$ equal to $0$ .

${(x + 5)}^{2} = 0$

Step 4.2.3.2

Solve ${(x + 5)}^{2} = 0$ for $x$ .

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Step 4.2.3.2.1

Set the $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 4.2.3.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4.2.4

Set ${(x - 5)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.2.4.1

Set ${(x - 5)}^{2}$ equal to $0$ .

${(x - 5)}^{2} = 0$

Step 4.2.4.2

Solve ${(x - 5)}^{2} = 0$ for $x$ .

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Step 4.2.4.2.1

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.2.4.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.2.5

The final solution is all the values that make ${(x + 5)}^{2} {(x - 5)}^{2} = 0$ true.

$x = - 5, 5$

Step 4.3

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

$x = - 5, x = 5$

Step 5

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 5) \cup (- 5, 0) \cup (0, 5) \cup (5, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 5)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 6$ in the expression.

$f' (- 6) = - \frac{4 (- 6)}{{({(- 6)}^{2} - 25)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $4$ by $- 6$ .

$f' (- 6) = - \frac{- 24}{{({(- 6)}^{2} - 25)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Raise $- 6$ to the power of $2$ .

$f' (- 6) = - \frac{- 24}{{(36 - 25)}^{2}}$

Step 6.2.2.2

Subtract $25$ from $36$ .

$f' (- 6) = - \frac{- 24}{11^{2}}$

Step 6.2.2.3

Raise $11$ to the power of $2$ .

$f' (- 6) = - \frac{- 24}{121}$

Step 6.2.3

Move the negative in front of the fraction.

$f' (- 6) = \frac{24}{121}$

Step 6.2.4

The final answer is $\frac{24}{121}$ .

$\frac{24}{121}$

Step 6.3

At $x = - 6$ the derivative is $\frac{24}{121}$ . Since this is positive, the function is increasing on $(- \infty, - 5)$ .

Increasing on $(- \infty, - 5)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(- 5, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- \frac{5}{2}$ in the expression.

$f' (- \frac{5}{2}) = - \frac{4 (- \frac{5}{2})}{{({(- \frac{5}{2})}^{2} - 25)}^{2}}$

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 $- 1$ by $4$ .

$f' (- \frac{5}{2}) = - \frac{- 4 (\frac{5}{2})}{{({(- \frac{5}{2})}^{2} - 25)}^{2}}$

Step 7.2.1.2

Combine $- 4$ and $\frac{5}{2}$ .

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

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.2.2.1.1

Apply the product rule to $- \frac{5}{2}$ .

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

Step 7.2.2.1.2

Apply the product rule to $\frac{5}{2}$ .

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

Step 7.2.2.2

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

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

Step 7.2.2.3

Multiply $\frac{5^{2}}{2^{2}}$ by $1$ .

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

Step 7.2.2.4

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

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

Step 7.2.2.5

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

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

Step 7.2.2.6

To write $- 25$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 7.2.2.7

Combine $- 25$ and $\frac{4}{4}$ .

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

Step 7.2.2.8

Combine the numerators over the common denominator.

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

Step 7.2.2.9

Simplify the numerator.

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Step 7.2.2.9.1

Multiply $- 25$ by $4$ .

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

Step 7.2.2.9.2

Subtract $100$ from $25$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 4 \cdot 5}{2}}{{(\frac{- 75}{4})}^{2}}$

Step 7.2.2.10

Move the negative in front of the fraction.

$f' (- \frac{5}{2}) = - \frac{\frac{- 4 \cdot 5}{2}}{{(- \frac{75}{4})}^{2}}$

Step 7.2.2.11

Apply the product rule to $- \frac{75}{4}$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 4 \cdot 5}{2}}{{(- 1)}^{2} {(\frac{75}{4})}^{2}}$

Step 7.2.2.12

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

$f' (- \frac{5}{2}) = - \frac{\frac{- 4 \cdot 5}{2}}{1 {(\frac{75}{4})}^{2}}$

Step 7.2.2.13

Apply the product rule to $\frac{75}{4}$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 4 \cdot 5}{2}}{1 (\frac{75^{2}}{4^{2}})}$

Step 7.2.2.14

Raise $75$ to the power of $2$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 4 \cdot 5}{2}}{1 (\frac{5625}{4^{2}})}$

Step 7.2.2.15

Raise $4$ to the power of $2$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 4 \cdot 5}{2}}{1 (\frac{5625}{16})}$

Step 7.2.2.16

Multiply $\frac{5625}{16}$ by $1$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 4 \cdot 5}{2}}{\frac{5625}{16}}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Multiply $- 4$ by $5$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 20}{2}}{\frac{5625}{16}}$

Step 7.2.3.2

Divide $- 20$ by $2$ .

$f' (- \frac{5}{2}) = - \frac{- 10}{\frac{5625}{16}}$

Step 7.2.4

Multiply the numerator by the reciprocal of the denominator.

$f' (- \frac{5}{2}) = - (- 10 (\frac{16}{5625}))$

Step 7.2.5

Cancel the common factor of $5$ .

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Step 7.2.5.1

Factor $5$ out of $- 10$ .

$f' (- \frac{5}{2}) = - (5 (- 2) (\frac{16}{5625}))$

Step 7.2.5.2

Factor $5$ out of $5625$ .

$f' (- \frac{5}{2}) = - (5 \cdot (- 2 \frac{16}{5 \cdot 1125}))$

Step 7.2.5.3

Cancel the common factor.

$f' (- \frac{5}{2}) = - (5 \cdot (- 2 \frac{16}{5 \cdot 1125}))$

Step 7.2.5.4

Rewrite the expression.

$f' (- \frac{5}{2}) = - (- 2 (\frac{16}{1125}))$

Step 7.2.6

Combine $- 2$ and $\frac{16}{1125}$ .

$f' (- \frac{5}{2}) = - \frac{- 2 \cdot 16}{1125}$

Step 7.2.7

Simplify the expression.

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Step 7.2.7.1

Multiply $- 2$ by $16$ .

$f' (- \frac{5}{2}) = - \frac{- 32}{1125}$

Step 7.2.7.2

Move the negative in front of the fraction.

$f' (- \frac{5}{2}) = \frac{32}{1125}$

Step 7.2.8

The final answer is $\frac{32}{1125}$ .

$\frac{32}{1125}$

Step 7.3

At $x = - \frac{5}{2}$ the derivative is $\frac{32}{1125}$ . Since this is positive, the function is increasing on $(- 5, 0)$ .

Increasing on $(- 5, 0)$ since $f' (x) > 0$

Step 8

Substitute a value from the interval $(0, 5)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $\frac{5}{2}$ in the expression.

$f' (\frac{5}{2}) = - \frac{4 (\frac{5}{2})}{{({(\frac{5}{2})}^{2} - 25)}^{2}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Combine $4$ and $\frac{5}{2}$ .

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

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Apply the product rule to $\frac{5}{2}$ .

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

Step 8.2.2.2

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

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

Step 8.2.2.3

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

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

Step 8.2.2.4

To write $- 25$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 8.2.2.5

Combine $- 25$ and $\frac{4}{4}$ .

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

Step 8.2.2.6

Combine the numerators over the common denominator.

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

Step 8.2.2.7

Simplify the numerator.

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Step 8.2.2.7.1

Multiply $- 25$ by $4$ .

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

Step 8.2.2.7.2

Subtract $100$ from $25$ .

$f' (\frac{5}{2}) = - \frac{\frac{4 \cdot 5}{2}}{{(\frac{- 75}{4})}^{2}}$

Step 8.2.2.8

Move the negative in front of the fraction.

$f' (\frac{5}{2}) = - \frac{\frac{4 \cdot 5}{2}}{{(- \frac{75}{4})}^{2}}$

Step 8.2.2.9

Apply the product rule to $- \frac{75}{4}$ .

$f' (\frac{5}{2}) = - \frac{\frac{4 \cdot 5}{2}}{{(- 1)}^{2} {(\frac{75}{4})}^{2}}$

Step 8.2.2.10

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

$f' (\frac{5}{2}) = - \frac{\frac{4 \cdot 5}{2}}{1 {(\frac{75}{4})}^{2}}$

Step 8.2.2.11

Apply the product rule to $\frac{75}{4}$ .

$f' (\frac{5}{2}) = - \frac{\frac{4 \cdot 5}{2}}{1 (\frac{75^{2}}{4^{2}})}$

Step 8.2.2.12

Raise $75$ to the power of $2$ .

$f' (\frac{5}{2}) = - \frac{\frac{4 \cdot 5}{2}}{1 (\frac{5625}{4^{2}})}$

Step 8.2.2.13

Raise $4$ to the power of $2$ .

$f' (\frac{5}{2}) = - \frac{\frac{4 \cdot 5}{2}}{1 (\frac{5625}{16})}$

Step 8.2.2.14

Multiply $\frac{5625}{16}$ by $1$ .

$f' (\frac{5}{2}) = - \frac{\frac{4 \cdot 5}{2}}{\frac{5625}{16}}$

Step 8.2.3

Simplify the expression.

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Step 8.2.3.1

Multiply $4$ by $5$ .

$f' (\frac{5}{2}) = - \frac{\frac{20}{2}}{\frac{5625}{16}}$

Step 8.2.3.2

Divide $20$ by $2$ .

$f' (\frac{5}{2}) = - \frac{10}{\frac{5625}{16}}$

Step 8.2.4

Multiply the numerator by the reciprocal of the denominator.

$f' (\frac{5}{2}) = - (10 (\frac{16}{5625}))$

Step 8.2.5

Cancel the common factor of $5$ .

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Step 8.2.5.1

Factor $5$ out of $10$ .

$f' (\frac{5}{2}) = - (5 (2) (\frac{16}{5625}))$

Step 8.2.5.2

Factor $5$ out of $5625$ .

$f' (\frac{5}{2}) = - (5 \cdot (2 (\frac{16}{5 \cdot 1125})))$

Step 8.2.5.3

Cancel the common factor.

$f' (\frac{5}{2}) = - (5 \cdot (2 (\frac{16}{5 \cdot 1125})))$

Step 8.2.5.4

Rewrite the expression.

$f' (\frac{5}{2}) = - (2 (\frac{16}{1125}))$

Step 8.2.6

Combine $2$ and $\frac{16}{1125}$ .

$f' (\frac{5}{2}) = - \frac{2 \cdot 16}{1125}$

Step 8.2.7

Multiply $2$ by $16$ .

$f' (\frac{5}{2}) = - \frac{32}{1125}$

Step 8.2.8

The final answer is $- \frac{32}{1125}$ .

$- \frac{32}{1125}$

Step 8.3

At $x = \frac{5}{2}$ the derivative is $- \frac{32}{1125}$ . Since this is negative, the function is decreasing on $(0, 5)$ .

Decreasing on $(0, 5)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(5, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $6$ in the expression.

$f' (6) = - \frac{4 (6)}{{({(6)}^{2} - 25)}^{2}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Multiply $4$ by $6$ .

$f' (6) = - \frac{24}{{(6^{2} - 25)}^{2}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Raise $6$ to the power of $2$ .

$f' (6) = - \frac{24}{{(36 - 25)}^{2}}$

Step 9.2.2.2

Subtract $25$ from $36$ .

$f' (6) = - \frac{24}{11^{2}}$

Step 9.2.2.3

Raise $11$ to the power of $2$ .

$f' (6) = - \frac{24}{121}$

Step 9.2.3

The final answer is $- \frac{24}{121}$ .

$- \frac{24}{121}$

Step 9.3

At $x = 6$ the derivative is $- \frac{24}{121}$ . Since this is negative, the function is decreasing on $(5, \infty)$ .

Decreasing on $(5, \infty)$ since $f' (x) < 0$

Step 10

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, - 5), (- 5, 0)$

Decreasing on: $(0, 5), (5, \infty)$

Step 11