Calculus Examples

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

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

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 x^{2}}{x - 3}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x^{2}}{x - 3}]$ .

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

Step 1.1.2

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 $f (x) = x^{2}$ and $g (x) = x - 3$ .

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

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

Step 1.1.3.2

Move $2$ to the left of $x - 3$ .

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

Step 1.1.3.3

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

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

Step 1.1.3.4

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

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

Step 1.1.3.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 1.1.3.6

Combine fractions.

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Step 1.1.3.6.1

Add $1$ and $0$ .

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

Step 1.1.3.6.2

Multiply $- 1$ by $1$ .

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

Step 1.1.3.6.3

Combine $4$ and $\frac{2 (x - 3) x - x^{2}}{{(x - 3)}^{2}}$ .

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

Apply the distributive property.

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

Step 1.1.4.2

Apply the distributive property.

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

Step 1.1.4.3

Apply the distributive property.

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

Step 1.1.4.4

Simplify the numerator.

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Step 1.1.4.4.1

Simplify each term.

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Step 1.1.4.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.4.4.1.1.1

Move $x$ .

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

Step 1.1.4.4.1.1.2

Multiply $x$ by $x$ .

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

Step 1.1.4.4.1.2

Multiply $2$ by $4$ .

$\frac{8 x^{2} + 4 (2 \cdot - 3 x) + 4 (- x^{2})}{{(x - 3)}^{2}}$

Step 1.1.4.4.1.3

Multiply $2$ by $- 3$ .

$\frac{8 x^{2} + 4 (- 6 x) + 4 (- x^{2})}{{(x - 3)}^{2}}$

Step 1.1.4.4.1.4

Multiply $- 6$ by $4$ .

$\frac{8 x^{2} - 24 x + 4 (- x^{2})}{{(x - 3)}^{2}}$

Step 1.1.4.4.1.5

Multiply $- 1$ by $4$ .

$\frac{8 x^{2} - 24 x - 4 x^{2}}{{(x - 3)}^{2}}$

Step 1.1.4.4.2

Subtract $4 x^{2}$ from $8 x^{2}$ .

$\frac{4 x^{2} - 24 x}{{(x - 3)}^{2}}$

Step 1.1.4.5

Factor $4 x$ out of $4 x^{2} - 24 x$ .

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Step 1.1.4.5.1

Factor $4 x$ out of $4 x^{2}$ .

$\frac{4 x (x) - 24 x}{{(x - 3)}^{2}}$

Step 1.1.4.5.2

Factor $4 x$ out of $- 24 x$ .

$\frac{4 x (x) + 4 x (- 6)}{{(x - 3)}^{2}}$

Step 1.1.4.5.3

Factor $4 x$ out of $4 x (x) + 4 x (- 6)$ .

$f' (x) = \frac{4 x (x - 6)}{{(x - 3)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{4 x (x - 6)}{{(x - 3)}^{2}}$ .

$\frac{4 x (x - 6)}{{(x - 3)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{4 x (x - 6)}{{(x - 3)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{4 x (x - 6)}{{(x - 3)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$4 x (x - 6) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

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

$x = 0$

$x - 6 = 0$

Step 2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.3.3

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 2.3.3.2

Add $6$ to both sides of the equation.

$x = 6$

Step 2.3.4

The final solution is all the values that make $4 x (x - 6) = 0$ true.

$x = 0, 6$

Step 3

The values which make the derivative equal to $0$ are $0, 6$ .

$0, 6$

Step 4

Find where the derivative is undefined.

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Step 4.1

Set the denominator in $\frac{4 x (x - 6)}{{(x - 3)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x - 3)}^{2} = 0$

Step 4.2

Solve for $x$ .

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Step 4.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5

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

$(- \infty, 0) \cup (0, 3) \cup (3, 6) \cup (6, \infty)$

Step 6

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 6.1

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

$f' (- 1) = \frac{4 (- 1) ((- 1) - 6)}{{((- 1) - 3)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $4$ by $- 1$ .

$f' (- 1) = \frac{- 4 (- 1 - 6)}{{(- 1 - 3)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Subtract $3$ from $- 1$ .

$f' (- 1) = \frac{- 4 (- 1 - 6)}{{(- 4)}^{2}}$

Step 6.2.2.2

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

$f' (- 1) = \frac{- 4 (- 1 - 6)}{16}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Subtract $6$ from $- 1$ .

$f' (- 1) = \frac{- 4 \cdot - 7}{16}$

Step 6.2.3.2

Multiply $- 4$ by $- 7$ .

$f' (- 1) = \frac{28}{16}$

Step 6.2.3.3

Cancel the common factor of $28$ and $16$ .

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Step 6.2.3.3.1

Factor $4$ out of $28$ .

$f' (- 1) = \frac{4 (7)}{16}$

Step 6.2.3.3.2

Cancel the common factors.

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Step 6.2.3.3.2.1

Factor $4$ out of $16$ .

$f' (- 1) = \frac{4 \cdot 7}{4 \cdot 4}$

Step 6.2.3.3.2.2

Cancel the common factor.

$f' (- 1) = \frac{4 \cdot 7}{4 \cdot 4}$

Step 6.2.3.3.2.3

Rewrite the expression.

$f' (- 1) = \frac{7}{4}$

Step 6.2.4

The final answer is $\frac{7}{4}$ .

$\frac{7}{4}$

Step 6.3

At $x = - 1$ the derivative is $\frac{7}{4}$ . Since this is positive, the function is increasing on $(- \infty, 0)$ .

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

Step 7

Substitute a value from the interval $(0, 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $\frac{3}{2}$ in the expression.

$f' (\frac{3}{2}) = \frac{4 (\frac{3}{2}) ((\frac{3}{2}) - 6)}{{((\frac{3}{2}) - 3)}^{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Combine $4$ and $\frac{3}{2}$ .

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{{(\frac{3}{2} - 3)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{{(\frac{3}{2} - 3 \cdot \frac{2}{2})}^{2}}$

Step 7.2.2.2

Combine $- 3$ and $\frac{2}{2}$ .

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{{(\frac{3}{2} + \frac{- 3 \cdot 2}{2})}^{2}}$

Step 7.2.2.3

Combine the numerators over the common denominator.

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{{(\frac{3 - 3 \cdot 2}{2})}^{2}}$

Step 7.2.2.4

Simplify the numerator.

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Step 7.2.2.4.1

Multiply $- 3$ by $2$ .

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{{(\frac{3 - 6}{2})}^{2}}$

Step 7.2.2.4.2

Subtract $6$ from $3$ .

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{{(\frac{- 3}{2})}^{2}}$

Step 7.2.2.5

Move the negative in front of the fraction.

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{{(- \frac{3}{2})}^{2}}$

Step 7.2.2.6

Apply the product rule to $- \frac{3}{2}$ .

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{{(- 1)}^{2} {(\frac{3}{2})}^{2}}$

Step 7.2.2.7

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

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{1 {(\frac{3}{2})}^{2}}$

Step 7.2.2.8

Apply the product rule to $\frac{3}{2}$ .

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{1 (\frac{3^{2}}{2^{2}})}$

Step 7.2.2.9

Raise $3$ to the power of $2$ .

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{1 (\frac{9}{2^{2}})}$

Step 7.2.2.10

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

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{1 (\frac{9}{4})}$

Step 7.2.2.11

Multiply $\frac{9}{4}$ by $1$ .

$f' (\frac{3}{2}) = \frac{\frac{4 \cdot 3}{2} \cdot (\frac{3}{2} - 6)}{\frac{9}{4}}$

Step 7.2.3

Multiply $4$ by $3$ .

$f' (\frac{3}{2}) = \frac{\frac{12}{2} \cdot (\frac{3}{2} - 6)}{\frac{9}{4}}$

Step 7.2.4

Simplify the numerator.

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Step 7.2.4.1

Divide $12$ by $2$ .

$f' (\frac{3}{2}) = \frac{6 (\frac{3}{2} - 6)}{\frac{9}{4}}$

Step 7.2.4.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{3}{2}) = \frac{6 (\frac{3}{2} - 6 \cdot \frac{2}{2})}{\frac{9}{4}}$

Step 7.2.4.3

Combine $- 6$ and $\frac{2}{2}$ .

$f' (\frac{3}{2}) = \frac{6 (\frac{3}{2} + \frac{- 6 \cdot 2}{2})}{\frac{9}{4}}$

Step 7.2.4.4

Combine the numerators over the common denominator.

$f' (\frac{3}{2}) = \frac{6 (\frac{3 - 6 \cdot 2}{2})}{\frac{9}{4}}$

Step 7.2.4.5

Simplify the numerator.

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Step 7.2.4.5.1

Multiply $- 6$ by $2$ .

$f' (\frac{3}{2}) = \frac{6 (\frac{3 - 12}{2})}{\frac{9}{4}}$

Step 7.2.4.5.2

Subtract $12$ from $3$ .

$f' (\frac{3}{2}) = \frac{6 (\frac{- 9}{2})}{\frac{9}{4}}$

Step 7.2.4.6

Move the negative in front of the fraction.

$f' (\frac{3}{2}) = \frac{6 \cdot (- 1 (\frac{9}{2}))}{\frac{9}{4}}$

Step 7.2.4.7

Combine exponents.

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Step 7.2.4.7.1

Factor out negative.

$f' (\frac{3}{2}) = \frac{- (6 (\frac{9}{2}))}{\frac{9}{4}}$

Step 7.2.4.7.2

Combine $6$ and $\frac{9}{2}$ .

$f' (\frac{3}{2}) = \frac{- \frac{6 \cdot 9}{2}}{\frac{9}{4}}$

Step 7.2.4.7.3

Multiply $6$ by $9$ .

$f' (\frac{3}{2}) = \frac{- \frac{54}{2}}{\frac{9}{4}}$

Step 7.2.4.8

Divide $54$ by $2$ .

$f' (\frac{3}{2}) = \frac{- 1 \cdot 27}{\frac{9}{4}}$

Step 7.2.5

Multiply $- 1$ by $27$ .

$f' (\frac{3}{2}) = \frac{- 27}{\frac{9}{4}}$

Step 7.2.6

Multiply the numerator by the reciprocal of the denominator.

$f' (\frac{3}{2}) = - 27 (\frac{4}{9})$

Step 7.2.7

Cancel the common factor of $9$ .

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Step 7.2.7.1

Factor $9$ out of $- 27$ .

$f' (\frac{3}{2}) = 9 (- 3) (\frac{4}{9})$

Step 7.2.7.2

Cancel the common factor.

$f' (\frac{3}{2}) = 9 \cdot (- 3 (\frac{4}{9}))$

Step 7.2.7.3

Rewrite the expression.

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

Step 7.2.8

Multiply $- 3$ by $4$ .

$f' (\frac{3}{2}) = - 12$

Step 7.2.9

The final answer is $- 12$ .

$- 12$

Step 7.3

At $x = \frac{3}{2}$ the derivative is $- 12$ . Since this is negative, the function is decreasing on $(0, 3)$ .

Decreasing on $(0, 3)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(3, 6)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $\frac{9}{2}$ in the expression.

$f' (\frac{9}{2}) = \frac{4 (\frac{9}{2}) ((\frac{9}{2}) - 6)}{{((\frac{9}{2}) - 3)}^{2}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Combine $4$ and $\frac{9}{2}$ .

$f' (\frac{9}{2}) = \frac{\frac{4 \cdot 9}{2} \cdot (\frac{9}{2} - 6)}{{(\frac{9}{2} - 3)}^{2}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{9}{2}) = \frac{\frac{4 \cdot 9}{2} \cdot (\frac{9}{2} - 6)}{{(\frac{9}{2} - 3 \cdot \frac{2}{2})}^{2}}$

Step 8.2.2.2

Combine $- 3$ and $\frac{2}{2}$ .

$f' (\frac{9}{2}) = \frac{\frac{4 \cdot 9}{2} \cdot (\frac{9}{2} - 6)}{{(\frac{9}{2} + \frac{- 3 \cdot 2}{2})}^{2}}$

Step 8.2.2.3

Combine the numerators over the common denominator.

$f' (\frac{9}{2}) = \frac{\frac{4 \cdot 9}{2} \cdot (\frac{9}{2} - 6)}{{(\frac{9 - 3 \cdot 2}{2})}^{2}}$

Step 8.2.2.4

Simplify the numerator.

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Step 8.2.2.4.1

Multiply $- 3$ by $2$ .

$f' (\frac{9}{2}) = \frac{\frac{4 \cdot 9}{2} \cdot (\frac{9}{2} - 6)}{{(\frac{9 - 6}{2})}^{2}}$

Step 8.2.2.4.2

Subtract $6$ from $9$ .

$f' (\frac{9}{2}) = \frac{\frac{4 \cdot 9}{2} \cdot (\frac{9}{2} - 6)}{{(\frac{3}{2})}^{2}}$

Step 8.2.2.5

Apply the product rule to $\frac{3}{2}$ .

$f' (\frac{9}{2}) = \frac{\frac{4 \cdot 9}{2} \cdot (\frac{9}{2} - 6)}{\frac{3^{2}}{2^{2}}}$

Step 8.2.2.6

Raise $3$ to the power of $2$ .

$f' (\frac{9}{2}) = \frac{\frac{4 \cdot 9}{2} \cdot (\frac{9}{2} - 6)}{\frac{9}{2^{2}}}$

Step 8.2.2.7

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

$f' (\frac{9}{2}) = \frac{\frac{4 \cdot 9}{2} \cdot (\frac{9}{2} - 6)}{\frac{9}{4}}$

Step 8.2.3

Multiply $4$ by $9$ .

$f' (\frac{9}{2}) = \frac{\frac{36}{2} \cdot (\frac{9}{2} - 6)}{\frac{9}{4}}$

Step 8.2.4

Simplify the numerator.

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Step 8.2.4.1

Divide $36$ by $2$ .

$f' (\frac{9}{2}) = \frac{18 (\frac{9}{2} - 6)}{\frac{9}{4}}$

Step 8.2.4.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{9}{2}) = \frac{18 (\frac{9}{2} - 6 \cdot \frac{2}{2})}{\frac{9}{4}}$

Step 8.2.4.3

Combine $- 6$ and $\frac{2}{2}$ .

$f' (\frac{9}{2}) = \frac{18 (\frac{9}{2} + \frac{- 6 \cdot 2}{2})}{\frac{9}{4}}$

Step 8.2.4.4

Combine the numerators over the common denominator.

$f' (\frac{9}{2}) = \frac{18 (\frac{9 - 6 \cdot 2}{2})}{\frac{9}{4}}$

Step 8.2.4.5

Simplify the numerator.

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Step 8.2.4.5.1

Multiply $- 6$ by $2$ .

$f' (\frac{9}{2}) = \frac{18 (\frac{9 - 12}{2})}{\frac{9}{4}}$

Step 8.2.4.5.2

Subtract $12$ from $9$ .

$f' (\frac{9}{2}) = \frac{18 (\frac{- 3}{2})}{\frac{9}{4}}$

Step 8.2.4.6

Move the negative in front of the fraction.

$f' (\frac{9}{2}) = \frac{18 \cdot (- 1 (\frac{3}{2}))}{\frac{9}{4}}$

Step 8.2.4.7

Combine exponents.

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Step 8.2.4.7.1

Factor out negative.

$f' (\frac{9}{2}) = \frac{- (18 (\frac{3}{2}))}{\frac{9}{4}}$

Step 8.2.4.7.2

Combine $18$ and $\frac{3}{2}$ .

$f' (\frac{9}{2}) = \frac{- \frac{18 \cdot 3}{2}}{\frac{9}{4}}$

Step 8.2.4.7.3

Multiply $18$ by $3$ .

$f' (\frac{9}{2}) = \frac{- \frac{54}{2}}{\frac{9}{4}}$

Step 8.2.4.8

Divide $54$ by $2$ .

$f' (\frac{9}{2}) = \frac{- 1 \cdot 27}{\frac{9}{4}}$

Step 8.2.5

Multiply $- 1$ by $27$ .

$f' (\frac{9}{2}) = \frac{- 27}{\frac{9}{4}}$

Step 8.2.6

Multiply the numerator by the reciprocal of the denominator.

$f' (\frac{9}{2}) = - 27 (\frac{4}{9})$

Step 8.2.7

Cancel the common factor of $9$ .

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Step 8.2.7.1

Factor $9$ out of $- 27$ .

$f' (\frac{9}{2}) = 9 (- 3) (\frac{4}{9})$

Step 8.2.7.2

Cancel the common factor.

$f' (\frac{9}{2}) = 9 \cdot (- 3 (\frac{4}{9}))$

Step 8.2.7.3

Rewrite the expression.

$f' (\frac{9}{2}) = - 3 \cdot 4$

Step 8.2.8

Multiply $- 3$ by $4$ .

$f' (\frac{9}{2}) = - 12$

Step 8.2.9

The final answer is $- 12$ .

$- 12$

Step 8.3

At $x = \frac{9}{2}$ the derivative is $- 12$ . Since this is negative, the function is decreasing on $(3, 6)$ .

Decreasing on $(3, 6)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(6, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $7$ in the expression.

$f' (7) = \frac{4 (7) ((7) - 6)}{{((7) - 3)}^{2}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Multiply $4$ by $7$ .

$f' (7) = \frac{28 (7 - 6)}{{(7 - 3)}^{2}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Subtract $3$ from $7$ .

$f' (7) = \frac{28 (7 - 6)}{4^{2}}$

Step 9.2.2.2

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

$f' (7) = \frac{28 (7 - 6)}{16}$

Step 9.2.3

Reduce the expression by cancelling the common factors.

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Step 9.2.3.1

Subtract $6$ from $7$ .

$f' (7) = \frac{28 \cdot 1}{16}$

Step 9.2.3.2

Multiply $28$ by $1$ .

$f' (7) = \frac{28}{16}$

Step 9.2.3.3

Cancel the common factor of $28$ and $16$ .

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Step 9.2.3.3.1

Factor $4$ out of $28$ .

$f' (7) = \frac{4 (7)}{16}$

Step 9.2.3.3.2

Cancel the common factors.

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Step 9.2.3.3.2.1

Factor $4$ out of $16$ .

$f' (7) = \frac{4 \cdot 7}{4 \cdot 4}$

Step 9.2.3.3.2.2

Cancel the common factor.

$f' (7) = \frac{4 \cdot 7}{4 \cdot 4}$

Step 9.2.3.3.2.3

Rewrite the expression.

$f' (7) = \frac{7}{4}$

Step 9.2.4

The final answer is $\frac{7}{4}$ .

$\frac{7}{4}$

Step 9.3

At $x = 7$ the derivative is $\frac{7}{4}$ . Since this is positive, the function is increasing on $(6, \infty)$ .

Increasing on $(6, \infty)$ since $f' (x) > 0$

Step 10

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

Increasing on: $(- \infty, 0), (6, \infty)$

Decreasing on: $(0, 3), (3, 6)$

Step 11