Calculus Examples

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

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

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

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 $n = 2$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

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 $n = 2$ .

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

Step 1.1.2.5

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

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

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $2 x$ and $0$ .

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

Step 1.1.2.6.2

Multiply $2$ by $- 1$ .

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

Add $1$ and $2$ .

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

Step 1.1.6

Simplify.

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Step 1.1.6.1

Apply the distributive property.

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

Step 1.1.6.2

Apply the distributive property.

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

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 $x$ by adding the exponents.

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Step 1.1.6.3.1.1.1

Move $x$ .

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

Step 1.1.6.3.1.1.2

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

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Step 1.1.6.3.1.1.2.1

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

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

Step 1.1.6.3.1.1.2.2

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

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

Step 1.1.6.3.1.1.3

Add $1$ and $2$ .

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

Step 1.1.6.3.1.2

Multiply $2$ by $- 25$ .

$f' (x) = \frac{2 x^{3} - 50 x - 2 x^{3}}{{(x^{2} - 25)}^{2}}$

Step 1.1.6.3.2

Combine the opposite terms in $2 x^{3} - 50 x - 2 x^{3}$ .

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Step 1.1.6.3.2.1

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

$f' (x) = \frac{- 50 x + 0}{{(x^{2} - 25)}^{2}}$

Step 1.1.6.3.2.2

Add $- 50 x$ and $0$ .

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

Step 1.1.6.4

Move the negative in front of the fraction.

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

Step 1.1.6.5

Simplify the denominator.

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Step 1.1.6.5.1

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

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

Step 1.1.6.5.2

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 $b = 5$ .

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

Step 1.1.6.5.3

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

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{50 x}{{(x + 5)}^{2} {(x - 5)}^{2}}$ .

$- \frac{50 x}{{(x + 5)}^{2} {(x - 5)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{50 x}{{(x + 5)}^{2} {(x - 5)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{50 x}{{(x + 5)}^{2} {(x - 5)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$50 x = 0$

Step 2.3

Divide each term in $50 x = 0$ by $50$ and simplify.

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Step 2.3.1

Divide each term in $50 x = 0$ by $50$ .

$\frac{50 x}{50} = \frac{0}{50}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $50$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{50 x}{50} = \frac{0}{50}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{50}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $50$ .

$x = 0$

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 $\frac{50 x}{{(x + 5)}^{2} {(x - 5)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 4.2

Solve for $x$ .

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Step 4.2.1

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

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

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

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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Step 4.2.2.2.1

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

$x + 5 = 0$

Step 4.2.2.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

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

Add $5$ to both sides of the equation.

$x = 5$

Step 4.2.4

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.

$f' (- 6) = - \frac{50 (- 6)}{{((- 6) + 5)}^{2} {((- 6) - 5)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $50$ by $- 6$ .

$f' (- 6) = - \frac{- 300}{{(- 6 + 5)}^{2} {(- 6 - 5)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Add $- 6$ and $5$ .

$f' (- 6) = - \frac{- 300}{{(- 1)}^{2} {(- 6 - 5)}^{2}}$

Step 6.2.2.2

Subtract $5$ from $- 6$ .

$f' (- 6) = - \frac{- 300}{{(- 1)}^{2} {(- 11)}^{2}}$

Step 6.2.2.3

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

$f' (- 6) = - \frac{- 300}{1 {(- 11)}^{2}}$

Step 6.2.2.4

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

$f' (- 6) = - \frac{- 300}{1 \cdot 121}$

Step 6.2.2.5

Multiply $121$ by $1$ .

$f' (- 6) = - \frac{- 300}{121}$

Step 6.2.3

Move the negative in front of the fraction.

$f' (- 6) = \frac{300}{121}$

Step 6.2.4

The final answer is $\frac{300}{121}$ .

$\frac{300}{121}$

Step 6.3

At $x = - 6$ the derivative is $\frac{300}{121}$ . 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.

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

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 $50$ .

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

Step 7.2.1.2

Combine $- 50$ and $\frac{5}{2}$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{{(- \frac{5}{2} + 5)}^{2} {(- \frac{5}{2} - 5)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{{(- \frac{5}{2} + 5 \cdot \frac{2}{2})}^{2} {(- \frac{5}{2} - 5)}^{2}}$

Step 7.2.2.2

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

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{{(- \frac{5}{2} + \frac{5 \cdot 2}{2})}^{2} {(- \frac{5}{2} - 5)}^{2}}$

Step 7.2.2.3

Combine the numerators over the common denominator.

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{{(\frac{- 5 + 5 \cdot 2}{2})}^{2} {(- \frac{5}{2} - 5)}^{2}}$

Step 7.2.2.4

Simplify the numerator.

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Step 7.2.2.4.1

Multiply $5$ by $2$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{{(\frac{- 5 + 10}{2})}^{2} {(- \frac{5}{2} - 5)}^{2}}$

Step 7.2.2.4.2

Add $- 5$ and $10$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{{(\frac{5}{2})}^{2} {(- \frac{5}{2} - 5)}^{2}}$

Step 7.2.2.5

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

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5^{2}}{2^{2}} \cdot {(- \frac{5}{2} - 5)}^{2}}$

Step 7.2.2.6

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5^{2}}{2^{2}} \cdot {(- \frac{5}{2} - 5 \cdot \frac{2}{2})}^{2}}$

Step 7.2.2.7

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

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5^{2}}{2^{2}} \cdot {(- \frac{5}{2} + \frac{- 5 \cdot 2}{2})}^{2}}$

Step 7.2.2.8

Combine the numerators over the common denominator.

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5^{2}}{2^{2}} \cdot {(\frac{- 5 - 5 \cdot 2}{2})}^{2}}$

Step 7.2.2.9

Simplify the numerator.

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Step 7.2.2.9.1

Multiply $- 5$ by $2$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5^{2}}{2^{2}} \cdot {(\frac{- 5 - 10}{2})}^{2}}$

Step 7.2.2.9.2

Subtract $10$ from $- 5$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5^{2}}{2^{2}} \cdot {(\frac{- 15}{2})}^{2}}$

Step 7.2.2.10

Move the negative in front of the fraction.

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5^{2}}{2^{2}} \cdot {(- \frac{15}{2})}^{2}}$

Step 7.2.2.11

Apply the product rule to $- \frac{15}{2}$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5^{2}}{2^{2}} \cdot ({(- 1)}^{2} {(\frac{15}{2})}^{2})}$

Step 7.2.2.12

Combine exponents.

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Step 7.2.2.12.1

Combine $\frac{5^{2}}{2^{2}}$ and ${(- 1)}^{2}$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5^{2} {(- 1)}^{2}}{2^{2}} \cdot {(\frac{15}{2})}^{2}}$

Step 7.2.2.12.2

Combine $\frac{5^{2} {(- 1)}^{2}}{2^{2}}$ and ${(\frac{15}{2})}^{2}$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5^{2} {(- 1)}^{2} {(\frac{15}{2})}^{2}}{2^{2}}}$

Step 7.2.2.13

Simplify the numerator.

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Step 7.2.2.13.1

Apply the product rule to $\frac{15}{2}$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5^{2} {(- 1)}^{2} (\frac{15^{2}}{2^{2}})}{2^{2}}}$

Step 7.2.2.13.2

Combine exponents.

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Step 7.2.2.13.2.1

Combine $5^{2}$ and $\frac{15^{2}}{2^{2}}$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{{(- 1)}^{2} (\frac{5^{2} \cdot 15^{2}}{2^{2}})}{2^{2}}}$

Step 7.2.2.13.2.2

Combine ${(- 1)}^{2}$ and $\frac{5^{2} \cdot 15^{2}}{2^{2}}$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{\frac{{(- 1)}^{2} (5^{2} \cdot 15^{2})}{2^{2}}}{2^{2}}}$

Step 7.2.2.13.3

Simplify the numerator.

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Step 7.2.2.13.3.1

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

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{\frac{1 \cdot 5^{2} \cdot 15^{2}}{2^{2}}}{2^{2}}}$

Step 7.2.2.13.3.2

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

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{\frac{1 \cdot 25 \cdot 15^{2}}{2^{2}}}{2^{2}}}$

Step 7.2.2.13.3.3

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

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{\frac{1 \cdot 25 \cdot 225}{2^{2}}}{2^{2}}}$

Step 7.2.2.13.3.4

Combine exponents.

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Step 7.2.2.13.3.4.1

Multiply $25$ by $1$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{\frac{25 \cdot 225}{2^{2}}}{2^{2}}}$

Step 7.2.2.13.3.4.2

Multiply $25$ by $225$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{\frac{5625}{2^{2}}}{2^{2}}}$

Step 7.2.2.13.4

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

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{\frac{5625}{4}}{2^{2}}}$

Step 7.2.2.14

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

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{\frac{5625}{4}}{4}}$

Step 7.2.2.15

Multiply the numerator by the reciprocal of the denominator.

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5625}{4} \cdot \frac{1}{4}}$

Step 7.2.2.16

Multiply $\frac{5625}{4} \cdot \frac{1}{4}$ .

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Step 7.2.2.16.1

Multiply $\frac{5625}{4}$ by $\frac{1}{4}$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5625}{4 \cdot 4}}$

Step 7.2.2.16.2

Multiply $4$ by $4$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 50 \cdot 5}{2}}{\frac{5625}{16}}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Multiply $- 50$ by $5$ .

$f' (- \frac{5}{2}) = - \frac{\frac{- 250}{2}}{\frac{5625}{16}}$

Step 7.2.3.2

Divide $- 250$ by $2$ .

$f' (- \frac{5}{2}) = - \frac{- 125}{\frac{5625}{16}}$

Step 7.2.4

Multiply the numerator by the reciprocal of the denominator.

$f' (- \frac{5}{2}) = - (- 125 (\frac{16}{5625}))$

Step 7.2.5

Cancel the common factor of $125$ .

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Step 7.2.5.1

Factor $125$ out of $- 125$ .

$f' (- \frac{5}{2}) = - (125 (- 1) (\frac{16}{5625}))$

Step 7.2.5.2

Factor $125$ out of $5625$ .

$f' (- \frac{5}{2}) = - (125 \cdot (- 1 \frac{16}{125 \cdot 45}))$

Step 7.2.5.3

Cancel the common factor.

$f' (- \frac{5}{2}) = - (125 \cdot (- 1 \frac{16}{125 \cdot 45}))$

Step 7.2.5.4

Rewrite the expression.

$f' (- \frac{5}{2}) = \frac{16}{45}$

Step 7.2.6

The final answer is $\frac{16}{45}$ .

$\frac{16}{45}$

Step 7.3

At $x = - \frac{5}{2}$ the derivative is $\frac{16}{45}$ . 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.

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

Step 8.2

Simplify the result.

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Step 8.2.1

Combine $50$ and $\frac{5}{2}$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{{(\frac{5}{2} + 5)}^{2} {(\frac{5}{2} - 5)}^{2}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{{(\frac{5}{2} + 5 \cdot \frac{2}{2})}^{2} {(\frac{5}{2} - 5)}^{2}}$

Step 8.2.2.2

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

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{{(\frac{5}{2} + \frac{5 \cdot 2}{2})}^{2} {(\frac{5}{2} - 5)}^{2}}$

Step 8.2.2.3

Combine the numerators over the common denominator.

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{{(\frac{5 + 5 \cdot 2}{2})}^{2} {(\frac{5}{2} - 5)}^{2}}$

Step 8.2.2.4

Simplify the numerator.

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Step 8.2.2.4.1

Multiply $5$ by $2$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{{(\frac{5 + 10}{2})}^{2} {(\frac{5}{2} - 5)}^{2}}$

Step 8.2.2.4.2

Add $5$ and $10$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{{(\frac{15}{2})}^{2} {(\frac{5}{2} - 5)}^{2}}$

Step 8.2.2.5

Apply the product rule to $\frac{15}{2}$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{15^{2}}{2^{2}} \cdot {(\frac{5}{2} - 5)}^{2}}$

Step 8.2.2.6

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{15^{2}}{2^{2}} \cdot {(\frac{5}{2} - 5 \cdot \frac{2}{2})}^{2}}$

Step 8.2.2.7

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

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

Step 8.2.2.8

Combine the numerators over the common denominator.

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{15^{2}}{2^{2}} \cdot {(\frac{5 - 5 \cdot 2}{2})}^{2}}$

Step 8.2.2.9

Simplify the numerator.

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Step 8.2.2.9.1

Multiply $- 5$ by $2$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{15^{2}}{2^{2}} \cdot {(\frac{5 - 10}{2})}^{2}}$

Step 8.2.2.9.2

Subtract $10$ from $5$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{15^{2}}{2^{2}} \cdot {(\frac{- 5}{2})}^{2}}$

Step 8.2.2.10

Move the negative in front of the fraction.

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{15^{2}}{2^{2}} \cdot {(- \frac{5}{2})}^{2}}$

Step 8.2.2.11

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

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{15^{2}}{2^{2}} \cdot ({(- 1)}^{2} {(\frac{5}{2})}^{2})}$

Step 8.2.2.12

Combine exponents.

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Step 8.2.2.12.1

Combine $\frac{15^{2}}{2^{2}}$ and ${(- 1)}^{2}$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{15^{2} {(- 1)}^{2}}{2^{2}} \cdot {(\frac{5}{2})}^{2}}$

Step 8.2.2.12.2

Combine $\frac{15^{2} {(- 1)}^{2}}{2^{2}}$ and ${(\frac{5}{2})}^{2}$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{15^{2} {(- 1)}^{2} {(\frac{5}{2})}^{2}}{2^{2}}}$

Step 8.2.2.13

Simplify the numerator.

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Step 8.2.2.13.1

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

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{15^{2} {(- 1)}^{2} (\frac{5^{2}}{2^{2}})}{2^{2}}}$

Step 8.2.2.13.2

Combine exponents.

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Step 8.2.2.13.2.1

Combine $15^{2}$ and $\frac{5^{2}}{2^{2}}$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{{(- 1)}^{2} (\frac{15^{2} \cdot 5^{2}}{2^{2}})}{2^{2}}}$

Step 8.2.2.13.2.2

Combine ${(- 1)}^{2}$ and $\frac{15^{2} \cdot 5^{2}}{2^{2}}$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{\frac{{(- 1)}^{2} (15^{2} \cdot 5^{2})}{2^{2}}}{2^{2}}}$

Step 8.2.2.13.3

Simplify the numerator.

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Step 8.2.2.13.3.1

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

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{\frac{1 \cdot 15^{2} \cdot 5^{2}}{2^{2}}}{2^{2}}}$

Step 8.2.2.13.3.2

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

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{\frac{1 \cdot 225 \cdot 5^{2}}{2^{2}}}{2^{2}}}$

Step 8.2.2.13.3.3

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

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{\frac{1 \cdot 225 \cdot 25}{2^{2}}}{2^{2}}}$

Step 8.2.2.13.3.4

Combine exponents.

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Step 8.2.2.13.3.4.1

Multiply $225$ by $1$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{\frac{225 \cdot 25}{2^{2}}}{2^{2}}}$

Step 8.2.2.13.3.4.2

Multiply $225$ by $25$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{\frac{5625}{2^{2}}}{2^{2}}}$

Step 8.2.2.13.4

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

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{\frac{5625}{4}}{2^{2}}}$

Step 8.2.2.14

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

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{\frac{5625}{4}}{4}}$

Step 8.2.2.15

Multiply the numerator by the reciprocal of the denominator.

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{5625}{4} \cdot \frac{1}{4}}$

Step 8.2.2.16

Multiply $\frac{5625}{4} \cdot \frac{1}{4}$ .

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Step 8.2.2.16.1

Multiply $\frac{5625}{4}$ by $\frac{1}{4}$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{5625}{4 \cdot 4}}$

Step 8.2.2.16.2

Multiply $4$ by $4$ .

$f' (\frac{5}{2}) = - \frac{\frac{50 \cdot 5}{2}}{\frac{5625}{16}}$

Step 8.2.3

Simplify the expression.

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Step 8.2.3.1

Multiply $50$ by $5$ .

$f' (\frac{5}{2}) = - \frac{\frac{250}{2}}{\frac{5625}{16}}$

Step 8.2.3.2

Divide $250$ by $2$ .

$f' (\frac{5}{2}) = - \frac{125}{\frac{5625}{16}}$

Step 8.2.4

Multiply the numerator by the reciprocal of the denominator.

$f' (\frac{5}{2}) = - (125 (\frac{16}{5625}))$

Step 8.2.5

Cancel the common factor of $125$ .

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Step 8.2.5.1

Factor $125$ out of $5625$ .

$f' (\frac{5}{2}) = - (125 (\frac{16}{125 (45)}))$

Step 8.2.5.2

Cancel the common factor.

$f' (\frac{5}{2}) = - (125 (\frac{16}{125 \cdot 45}))$

Step 8.2.5.3

Rewrite the expression.

$f' (\frac{5}{2}) = - \frac{16}{45}$

Step 8.2.6

The final answer is $- \frac{16}{45}$ .

$- \frac{16}{45}$

Step 8.3

At $x = \frac{5}{2}$ the derivative is $- \frac{16}{45}$ . 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.

$f' (6) = - \frac{50 (6)}{{((6) + 5)}^{2} {((6) - 5)}^{2}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Multiply $50$ by $6$ .

$f' (6) = - \frac{300}{{(6 + 5)}^{2} {(6 - 5)}^{2}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Add $6$ and $5$ .

$f' (6) = - \frac{300}{11^{2} {(6 - 5)}^{2}}$

Step 9.2.2.2

Subtract $5$ from $6$ .

$f' (6) = - \frac{300}{11^{2} \cdot 1^{2}}$

Step 9.2.2.3

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

$f' (6) = - \frac{300}{121 \cdot 1^{2}}$

Step 9.2.2.4

One to any power is one.

$f' (6) = - \frac{300}{121 \cdot 1}$

Step 9.2.3

Multiply $121$ by $1$ .

$f' (6) = - \frac{300}{121}$

Step 9.2.4

The final answer is $- \frac{300}{121}$ .

$- \frac{300}{121}$

Step 9.3

At $x = 6$ the derivative is $- \frac{300}{121}$ . 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