Calculus Examples

$\frac{x^{4}}{x - 15}$

Step 1

Write $\frac{x^{4}}{x - 15}$ as a function.

$f (x) = \frac{x^{4}}{x - 15}$

Step 2

Find the first 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 $f (x) = x^{4}$ and $g (x) = x - 15$ .

$\frac{(x - 15) \frac{d}{d x} [x^{4}] - x^{4} \frac{d}{d x} [x - 15]}{{(x - 15)}^{2}}$

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 $n = 4$ .

$\frac{(x - 15) (4 x^{3}) - x^{4} \frac{d}{d x} [x - 15]}{{(x - 15)}^{2}}$

Step 2.1.2.2

Move $4$ to the left of $x - 15$ .

$\frac{4 \cdot (x - 15) x^{3} - x^{4} \frac{d}{d x} [x - 15]}{{(x - 15)}^{2}}$

Step 2.1.2.3

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

$\frac{4 (x - 15) x^{3} - x^{4} (\frac{d}{d x} [x] + \frac{d}{d x} [- 15])}{{(x - 15)}^{2}}$

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 $n = 1$ .

$\frac{4 (x - 15) x^{3} - x^{4} (1 + \frac{d}{d x} [- 15])}{{(x - 15)}^{2}}$

Step 2.1.2.5

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

$\frac{4 (x - 15) x^{3} - x^{4} (1 + 0)}{{(x - 15)}^{2}}$

Step 2.1.2.6

Simplify the expression.

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Step 2.1.2.6.1

Add $1$ and $0$ .

$\frac{4 (x - 15) x^{3} - x^{4} \cdot 1}{{(x - 15)}^{2}}$

Step 2.1.2.6.2

Multiply $- 1$ by $1$ .

$\frac{4 (x - 15) x^{3} - x^{4}}{{(x - 15)}^{2}}$

Step 2.1.3

Simplify.

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Step 2.1.3.1

Apply the distributive property.

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

Step 2.1.3.2

Apply the distributive property.

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

Step 2.1.3.3

Simplify the numerator.

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Step 2.1.3.3.1

Simplify each term.

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Step 2.1.3.3.1.1

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

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Step 2.1.3.3.1.1.1

Move $x^{3}$ .

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

Step 2.1.3.3.1.1.2

Multiply $x^{3}$ by $x$ .

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Step 2.1.3.3.1.1.2.1

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

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

Step 2.1.3.3.1.1.2.2

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

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

Step 2.1.3.3.1.1.3

Add $3$ and $1$ .

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

Step 2.1.3.3.1.2

Multiply $4$ by $- 15$ .

$\frac{4 x^{4} - 60 x^{3} - x^{4}}{{(x - 15)}^{2}}$

Step 2.1.3.3.2

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

$\frac{3 x^{4} - 60 x^{3}}{{(x - 15)}^{2}}$

Step 2.1.3.4

Factor $3 x^{3}$ out of $3 x^{4} - 60 x^{3}$ .

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Step 2.1.3.4.1

Factor $3 x^{3}$ out of $3 x^{4}$ .

$\frac{3 x^{3} (x) - 60 x^{3}}{{(x - 15)}^{2}}$

Step 2.1.3.4.2

Factor $3 x^{3}$ out of $- 60 x^{3}$ .

$\frac{3 x^{3} (x) + 3 x^{3} (- 20)}{{(x - 15)}^{2}}$

Step 2.1.3.4.3

Factor $3 x^{3}$ out of $3 x^{3} (x) + 3 x^{3} (- 20)$ .

$f' (x) = \frac{3 x^{3} (x - 20)}{{(x - 15)}^{2}}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{3 x^{3} (x - 20)}{{(x - 15)}^{2}}$ .

$\frac{3 x^{3} (x - 20)}{{(x - 15)}^{2}}$

Step 3

Set the first derivative equal to $0$ then solve the equation $\frac{3 x^{3} (x - 20)}{{(x - 15)}^{2}} = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$\frac{3 x^{3} (x - 20)}{{(x - 15)}^{2}} = 0$

Step 3.2

Set the numerator equal to zero.

$3 x^{3} (x - 20) = 0$

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 $0$ .

$x^{3} = 0$

$x - 20 = 0$

Step 3.3.2

Set $x^{3}$ equal to $0$ and solve for $x$ .

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Step 3.3.2.1

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Step 3.3.2.2

Solve $x^{3} = 0$ for $x$ .

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Step 3.3.2.2.1

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

$x = \sqrt[3]{0}$

Step 3.3.2.2.2

Simplify $\sqrt[3]{0}$ .

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Step 3.3.2.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 3.3.2.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 3.3.3

Set $x - 20$ equal to $0$ and solve for $x$ .

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Step 3.3.3.1

Set $x - 20$ equal to $0$ .

$x - 20 = 0$

Step 3.3.3.2

Add $20$ to both sides of the equation.

$x = 20$

Step 3.3.4

The final solution is all the values that make $3 x^{3} (x - 20) = 0$ true.

$x = 0, 20$

Step 4

The values which make the derivative equal to $0$ are $0, 20$ .

$0, 20$

Step 5

Find where the derivative is undefined.

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Step 5.1

Set the denominator in $\frac{3 x^{3} (x - 20)}{{(x - 15)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 5.2

Solve for $x$ .

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Step 5.2.1

Set the $x - 15$ equal to $0$ .

$x - 15 = 0$

Step 5.2.2

Add $15$ to both sides of the equation.

$x = 15$

Step 6

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

$(- \infty, 0) \cup (0, 15) \cup (15, 20) \cup (20, \infty)$

Step 7

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

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Step 7.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = \frac{3 {(- 1)}^{3} ((- 1) - 20)}{{((- 1) - 15)}^{2}}$

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

Subtract $20$ from $- 1$ .

$f' (- 1) = \frac{3 {(- 1)}^{3} \cdot - 21}{{(- 1 - 15)}^{2}}$

Step 7.2.1.2

Multiply $- 21$ by $3$ .

$f' (- 1) = \frac{- 63 {(- 1)}^{3}}{{(- 1 - 15)}^{2}}$

Step 7.2.1.3

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

$f' (- 1) = \frac{- 63 \cdot - 1}{{(- 1 - 15)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Subtract $15$ from $- 1$ .

$f' (- 1) = \frac{- 63 \cdot - 1}{{(- 16)}^{2}}$

Step 7.2.2.2

Raise $- 16$ to the power of $2$ .

$f' (- 1) = \frac{- 63 \cdot - 1}{256}$

Step 7.2.3

Multiply $- 63$ by $- 1$ .

$f' (- 1) = \frac{63}{256}$

Step 7.2.4

The final answer is $\frac{63}{256}$ .

$\frac{63}{256}$

Step 7.3

At $x = - 1$ the derivative is $\frac{63}{256}$ . Since this is positive, the function is increasing on $(- \infty, 0)$ .

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

Step 8

Substitute a value from the interval $(0, 15)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $\frac{15}{2}$ in the expression.

$f' (\frac{15}{2}) = \frac{3 {(\frac{15}{2})}^{3} ((\frac{15}{2}) - 20)}{{((\frac{15}{2}) - 15)}^{2}}$

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

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

$f' (\frac{15}{2}) = \frac{3 (\frac{15^{3}}{2^{3}}) \cdot (\frac{15}{2} - 20)}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.2

Combine $3$ and $\frac{15^{3}}{2^{3}}$ .

$f' (\frac{15}{2}) = \frac{\frac{3 \cdot 15^{3}}{2^{3}} \cdot (\frac{15}{2} - 20)}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.3

To write $- 20$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{15}{2}) = \frac{\frac{3 \cdot 15^{3}}{2^{3}} \cdot (\frac{15}{2} - 20 \cdot \frac{2}{2})}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.4

Combine $- 20$ and $\frac{2}{2}$ .

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

Step 8.2.1.5

Combine the numerators over the common denominator.

$f' (\frac{15}{2}) = \frac{\frac{3 \cdot 15^{3}}{2^{3}} \cdot \frac{15 - 20 \cdot 2}{2}}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.6

Simplify the numerator.

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Step 8.2.1.6.1

Multiply $- 20$ by $2$ .

$f' (\frac{15}{2}) = \frac{\frac{3 \cdot 15^{3}}{2^{3}} \cdot \frac{15 - 40}{2}}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.6.2

Subtract $40$ from $15$ .

$f' (\frac{15}{2}) = \frac{\frac{3 \cdot 15^{3}}{2^{3}} \cdot \frac{- 25}{2}}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.7

Move the negative in front of the fraction.

$f' (\frac{15}{2}) = \frac{\frac{3 \cdot 15^{3}}{2^{3}} \cdot (- 1 (\frac{25}{2}))}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.8

Combine exponents.

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Step 8.2.1.8.1

Factor out negative.

$f' (\frac{15}{2}) = \frac{- (\frac{3 \cdot 15^{3}}{2^{3}} \cdot \frac{25}{2})}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.8.2

Multiply $\frac{3 \cdot 15^{3}}{2^{3}}$ by $\frac{25}{2}$ .

$f' (\frac{15}{2}) = \frac{- \frac{3 \cdot 15^{3} \cdot 25}{2^{3} \cdot 2}}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.8.3

Multiply $25$ by $3$ .

$f' (\frac{15}{2}) = \frac{- \frac{75 \cdot 15^{3}}{2^{3} \cdot 2}}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.8.4

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

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Step 8.2.1.8.4.1

Multiply $2^{3}$ by $2$ .

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Step 8.2.1.8.4.1.1

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

$f' (\frac{15}{2}) = \frac{- \frac{75 \cdot 15^{3}}{2^{3} \cdot 2}}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.8.4.1.2

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

$f' (\frac{15}{2}) = \frac{- \frac{75 \cdot 15^{3}}{2^{3 + 1}}}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.8.4.2

Add $3$ and $1$ .

$f' (\frac{15}{2}) = \frac{- \frac{75 \cdot 15^{3}}{2^{4}}}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.9

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

$f' (\frac{15}{2}) = \frac{- \frac{75 \cdot 3375}{2^{4}}}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.10

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

$f' (\frac{15}{2}) = \frac{- \frac{75 \cdot 3375}{16}}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.1.11

Multiply $75$ by $3375$ .

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{{(\frac{15}{2} - 15)}^{2}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

To write $- 15$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{{(\frac{15}{2} - 15 \cdot \frac{2}{2})}^{2}}$

Step 8.2.2.2

Combine $- 15$ and $\frac{2}{2}$ .

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{{(\frac{15}{2} + \frac{- 15 \cdot 2}{2})}^{2}}$

Step 8.2.2.3

Combine the numerators over the common denominator.

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{{(\frac{15 - 15 \cdot 2}{2})}^{2}}$

Step 8.2.2.4

Simplify the numerator.

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Step 8.2.2.4.1

Multiply $- 15$ by $2$ .

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{{(\frac{15 - 30}{2})}^{2}}$

Step 8.2.2.4.2

Subtract $30$ from $15$ .

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{{(\frac{- 15}{2})}^{2}}$

Step 8.2.2.5

Move the negative in front of the fraction.

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{{(- \frac{15}{2})}^{2}}$

Step 8.2.2.6

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

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{{(- 1)}^{2} {(\frac{15}{2})}^{2}}$

Step 8.2.2.7

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

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{1 {(\frac{15}{2})}^{2}}$

Step 8.2.2.8

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

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{1 (\frac{15^{2}}{2^{2}})}$

Step 8.2.2.9

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

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{1 (\frac{225}{2^{2}})}$

Step 8.2.2.10

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

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{1 (\frac{225}{4})}$

Step 8.2.2.11

Multiply $\frac{225}{4}$ by $1$ .

$f' (\frac{15}{2}) = \frac{- \frac{253125}{16}}{\frac{225}{4}}$

Step 8.2.3

Multiply the numerator by the reciprocal of the denominator.

$f' (\frac{15}{2}) = - \frac{253125}{16} \cdot \frac{4}{225}$

Step 8.2.4

Cancel the common factor of $225$ .

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Step 8.2.4.1

Move the leading negative in $- \frac{253125}{16}$ into the numerator.

$f' (\frac{15}{2}) = \frac{- 253125}{16} \cdot \frac{4}{225}$

Step 8.2.4.2

Factor $225$ out of $- 253125$ .

$f' (\frac{15}{2}) = \frac{225 (- 1125)}{16} \cdot \frac{4}{225}$

Step 8.2.4.3

Cancel the common factor.

$f' (\frac{15}{2}) = \frac{225 \cdot - 1125}{16} \cdot \frac{4}{225}$

Step 8.2.4.4

Rewrite the expression.

$f' (\frac{15}{2}) = \frac{- 1125}{16} \cdot 4$

Step 8.2.5

Cancel the common factor of $4$ .

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Step 8.2.5.1

Factor $4$ out of $16$ .

$f' (\frac{15}{2}) = \frac{- 1125}{4 (4)} \cdot 4$

Step 8.2.5.2

Cancel the common factor.

$f' (\frac{15}{2}) = \frac{- 1125}{4 \cdot 4} \cdot 4$

Step 8.2.5.3

Rewrite the expression.

$f' (\frac{15}{2}) = \frac{- 1125}{4}$

Step 8.2.6

Move the negative in front of the fraction.

$f' (\frac{15}{2}) = - \frac{1125}{4}$

Step 8.2.7

The final answer is $- \frac{1125}{4}$ .

$- \frac{1125}{4}$

Step 8.3

At $x = \frac{15}{2}$ the derivative is $- \frac{1125}{4}$ . Since this is negative, the function is decreasing on $(0, 15)$ .

Decreasing on $(0, 15)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(15, 20)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $\frac{35}{2}$ in the expression.

$f' (\frac{35}{2}) = \frac{3 {(\frac{35}{2})}^{3} ((\frac{35}{2}) - 20)}{{((\frac{35}{2}) - 15)}^{2}}$

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

Apply the product rule to $\frac{35}{2}$ .

$f' (\frac{35}{2}) = \frac{3 (\frac{35^{3}}{2^{3}}) \cdot (\frac{35}{2} - 20)}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.2

Combine $3$ and $\frac{35^{3}}{2^{3}}$ .

$f' (\frac{35}{2}) = \frac{\frac{3 \cdot 35^{3}}{2^{3}} \cdot (\frac{35}{2} - 20)}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.3

To write $- 20$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{35}{2}) = \frac{\frac{3 \cdot 35^{3}}{2^{3}} \cdot (\frac{35}{2} - 20 \cdot \frac{2}{2})}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.4

Combine $- 20$ and $\frac{2}{2}$ .

$f' (\frac{35}{2}) = \frac{\frac{3 \cdot 35^{3}}{2^{3}} \cdot (\frac{35}{2} + \frac{- 20 \cdot 2}{2})}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.5

Combine the numerators over the common denominator.

$f' (\frac{35}{2}) = \frac{\frac{3 \cdot 35^{3}}{2^{3}} \cdot \frac{35 - 20 \cdot 2}{2}}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.6

Simplify the numerator.

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Step 9.2.1.6.1

Multiply $- 20$ by $2$ .

$f' (\frac{35}{2}) = \frac{\frac{3 \cdot 35^{3}}{2^{3}} \cdot \frac{35 - 40}{2}}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.6.2

Subtract $40$ from $35$ .

$f' (\frac{35}{2}) = \frac{\frac{3 \cdot 35^{3}}{2^{3}} \cdot \frac{- 5}{2}}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.7

Move the negative in front of the fraction.

$f' (\frac{35}{2}) = \frac{\frac{3 \cdot 35^{3}}{2^{3}} \cdot (- 1 (\frac{5}{2}))}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.8

Combine exponents.

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Step 9.2.1.8.1

Factor out negative.

$f' (\frac{35}{2}) = \frac{- (\frac{3 \cdot 35^{3}}{2^{3}} \cdot \frac{5}{2})}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.8.2

Multiply $\frac{3 \cdot 35^{3}}{2^{3}}$ by $\frac{5}{2}$ .

$f' (\frac{35}{2}) = \frac{- \frac{3 \cdot 35^{3} \cdot 5}{2^{3} \cdot 2}}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.8.3

Multiply $5$ by $3$ .

$f' (\frac{35}{2}) = \frac{- \frac{15 \cdot 35^{3}}{2^{3} \cdot 2}}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.8.4

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

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Step 9.2.1.8.4.1

Multiply $2^{3}$ by $2$ .

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Step 9.2.1.8.4.1.1

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

$f' (\frac{35}{2}) = \frac{- \frac{15 \cdot 35^{3}}{2^{3} \cdot 2}}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.8.4.1.2

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

$f' (\frac{35}{2}) = \frac{- \frac{15 \cdot 35^{3}}{2^{3 + 1}}}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.8.4.2

Add $3$ and $1$ .

$f' (\frac{35}{2}) = \frac{- \frac{15 \cdot 35^{3}}{2^{4}}}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.9

Raise $35$ to the power of $3$ .

$f' (\frac{35}{2}) = \frac{- \frac{15 \cdot 42875}{2^{4}}}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.10

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

$f' (\frac{35}{2}) = \frac{- \frac{15 \cdot 42875}{16}}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.1.11

Multiply $15$ by $42875$ .

$f' (\frac{35}{2}) = \frac{- \frac{643125}{16}}{{(\frac{35}{2} - 15)}^{2}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

To write $- 15$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{35}{2}) = \frac{- \frac{643125}{16}}{{(\frac{35}{2} - 15 \cdot \frac{2}{2})}^{2}}$

Step 9.2.2.2

Combine $- 15$ and $\frac{2}{2}$ .

$f' (\frac{35}{2}) = \frac{- \frac{643125}{16}}{{(\frac{35}{2} + \frac{- 15 \cdot 2}{2})}^{2}}$

Step 9.2.2.3

Combine the numerators over the common denominator.

$f' (\frac{35}{2}) = \frac{- \frac{643125}{16}}{{(\frac{35 - 15 \cdot 2}{2})}^{2}}$

Step 9.2.2.4

Simplify the numerator.

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Step 9.2.2.4.1

Multiply $- 15$ by $2$ .

$f' (\frac{35}{2}) = \frac{- \frac{643125}{16}}{{(\frac{35 - 30}{2})}^{2}}$

Step 9.2.2.4.2

Subtract $30$ from $35$ .

$f' (\frac{35}{2}) = \frac{- \frac{643125}{16}}{{(\frac{5}{2})}^{2}}$

Step 9.2.2.5

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

$f' (\frac{35}{2}) = \frac{- \frac{643125}{16}}{\frac{5^{2}}{2^{2}}}$

Step 9.2.2.6

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

$f' (\frac{35}{2}) = \frac{- \frac{643125}{16}}{\frac{25}{2^{2}}}$

Step 9.2.2.7

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

$f' (\frac{35}{2}) = \frac{- \frac{643125}{16}}{\frac{25}{4}}$

Step 9.2.3

Multiply the numerator by the reciprocal of the denominator.

$f' (\frac{35}{2}) = - \frac{643125}{16} \cdot \frac{4}{25}$

Step 9.2.4

Cancel the common factor of $25$ .

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Step 9.2.4.1

Move the leading negative in $- \frac{643125}{16}$ into the numerator.

$f' (\frac{35}{2}) = \frac{- 643125}{16} \cdot \frac{4}{25}$

Step 9.2.4.2

Factor $25$ out of $- 643125$ .

$f' (\frac{35}{2}) = \frac{25 (- 25725)}{16} \cdot \frac{4}{25}$

Step 9.2.4.3

Cancel the common factor.

$f' (\frac{35}{2}) = \frac{25 \cdot - 25725}{16} \cdot \frac{4}{25}$

Step 9.2.4.4

Rewrite the expression.

$f' (\frac{35}{2}) = \frac{- 25725}{16} \cdot 4$

Step 9.2.5

Cancel the common factor of $4$ .

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Step 9.2.5.1

Factor $4$ out of $16$ .

$f' (\frac{35}{2}) = \frac{- 25725}{4 (4)} \cdot 4$

Step 9.2.5.2

Cancel the common factor.

$f' (\frac{35}{2}) = \frac{- 25725}{4 \cdot 4} \cdot 4$

Step 9.2.5.3

Rewrite the expression.

$f' (\frac{35}{2}) = \frac{- 25725}{4}$

Step 9.2.6

Move the negative in front of the fraction.

$f' (\frac{35}{2}) = - \frac{25725}{4}$

Step 9.2.7

The final answer is $- \frac{25725}{4}$ .

$- \frac{25725}{4}$

Step 9.3

At $x = \frac{35}{2}$ the derivative is $- \frac{25725}{4}$ . Since this is negative, the function is decreasing on $(15, 20)$ .

Decreasing on $(15, 20)$ since $f' (x) < 0$

Step 10

Substitute a value from the interval $(20, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 10.1

Replace the variable $x$ with $21$ in the expression.

$f' (21) = \frac{3 {(21)}^{3} ((21) - 20)}{{((21) - 15)}^{2}}$

Step 10.2

Simplify the result.

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Step 10.2.1

Simplify the numerator.

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Step 10.2.1.1

Subtract $20$ from $21$ .

$f' (21) = \frac{3 \cdot 21^{3} \cdot 1}{{(21 - 15)}^{2}}$

Step 10.2.1.2

Multiply $3$ by $1$ .

$f' (21) = \frac{3 \cdot 21^{3}}{{(21 - 15)}^{2}}$

Step 10.2.1.3

Raise $21$ to the power of $3$ .

$f' (21) = \frac{3 \cdot 9261}{{(21 - 15)}^{2}}$

Step 10.2.2

Simplify the denominator.

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Step 10.2.2.1

Subtract $15$ from $21$ .

$f' (21) = \frac{3 \cdot 9261}{6^{2}}$

Step 10.2.2.2

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

$f' (21) = \frac{3 \cdot 9261}{36}$

Step 10.2.3

Reduce the expression by cancelling the common factors.

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Step 10.2.3.1

Multiply $3$ by $9261$ .

$f' (21) = \frac{27783}{36}$

Step 10.2.3.2

Cancel the common factor of $27783$ and $36$ .

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Step 10.2.3.2.1

Factor $9$ out of $27783$ .

$f' (21) = \frac{9 (3087)}{36}$

Step 10.2.3.2.2

Cancel the common factors.

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Step 10.2.3.2.2.1

Factor $9$ out of $36$ .

$f' (21) = \frac{9 \cdot 3087}{9 \cdot 4}$

Step 10.2.3.2.2.2

Cancel the common factor.

$f' (21) = \frac{9 \cdot 3087}{9 \cdot 4}$

Step 10.2.3.2.2.3

Rewrite the expression.

$f' (21) = \frac{3087}{4}$

Step 10.2.4

The final answer is $\frac{3087}{4}$ .

$\frac{3087}{4}$

Step 10.3

At $x = 21$ the derivative is $\frac{3087}{4}$ . Since this is positive, the function is increasing on $(20, \infty)$ .

Increasing on $(20, \infty)$ since $f' (x) > 0$

Step 11

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

Increasing on: $(- \infty, 0), (20, \infty)$

Decreasing on: $(0, 15), (15, 20)$

Step 12