Calculus Examples

$\frac{x^{4}}{4} + 3 x^{3} + 9 x^{2}$

Step 1

Write $\frac{x^{4}}{4} + 3 x^{3} + 9 x^{2}$ as a function.

$f (x) = \frac{x^{4}}{4} + 3 x^{3} + 9 x^{2}$

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

By the Sum Rule, the derivative of $\frac{x^{4}}{4} + 3 x^{3} + 9 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{4}}{4}] + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [9 x^{2}]$ .

$\frac{d}{d x} [\frac{x^{4}}{4}] + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [9 x^{2}]$

Step 2.1.2

Evaluate $\frac{d}{d x} [\frac{x^{4}}{4}]$ .

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Step 2.1.2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x^{4}}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{4} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [9 x^{2}]$

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{4} (4 x^{3}) + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [9 x^{2}]$

Step 2.1.2.3

Combine $4$ and $\frac{1}{4}$ .

$\frac{4}{4} x^{3} + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [9 x^{2}]$

Step 2.1.2.4

Combine $\frac{4}{4}$ and $x^{3}$ .

$\frac{4 x^{3}}{4} + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [9 x^{2}]$

Step 2.1.2.5

Cancel the common factor of $4$ .

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Step 2.1.2.5.1

Cancel the common factor.

$\frac{4 x^{3}}{4} + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [9 x^{2}]$

Step 2.1.2.5.2

Divide $x^{3}$ by $1$ .

$x^{3} + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [9 x^{2}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [3 x^{3}]$ .

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Step 2.1.3.1

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

$x^{3} + 3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [9 x^{2}]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{3} + 3 (3 x^{2}) + \frac{d}{d x} [9 x^{2}]$

Step 2.1.3.3

Multiply $3$ by $3$ .

$x^{3} + 9 x^{2} + \frac{d}{d x} [9 x^{2}]$

Step 2.1.4

Evaluate $\frac{d}{d x} [9 x^{2}]$ .

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Step 2.1.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$x^{3} + 9 x^{2} + 9 \frac{d}{d x} [x^{2}]$

Step 2.1.4.2

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

$x^{3} + 9 x^{2} + 9 (2 x)$

Step 2.1.4.3

Multiply $2$ by $9$ .

$f' (x) = x^{3} + 9 x^{2} + 18 x$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $x^{3} + 9 x^{2} + 18 x$ .

$x^{3} + 9 x^{2} + 18 x$

Step 3

Set the first derivative equal to $0$ then solve the equation $x^{3} + 9 x^{2} + 18 x = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$x^{3} + 9 x^{2} + 18 x = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $x$ out of $x^{3} + 9 x^{2} + 18 x$ .

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Step 3.2.1.1

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

$x \cdot x^{2} + 9 x^{2} + 18 x = 0$

Step 3.2.1.2

Factor $x$ out of $9 x^{2}$ .

$x \cdot x^{2} + x (9 x) + 18 x = 0$

Step 3.2.1.3

Factor $x$ out of $18 x$ .

$x \cdot x^{2} + x (9 x) + x \cdot 18 = 0$

Step 3.2.1.4

Factor $x$ out of $x \cdot x^{2} + x (9 x)$ .

$x (x^{2} + 9 x) + x \cdot 18 = 0$

Step 3.2.1.5

Factor $x$ out of $x (x^{2} + 9 x) + x \cdot 18$ .

$x (x^{2} + 9 x + 18) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor $x^{2} + 9 x + 18$ using the AC method.

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Step 3.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $18$ and whose sum is $9$ .

$3, 6$

Step 3.2.2.1.2

Write the factored form using these integers.

$x ((x + 3) (x + 6)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$x (x + 3) (x + 6) = 0$

Step 3.3

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

$x = 0$

$x + 3 = 0$

$x + 6 = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.6

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

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Step 3.6.1

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

$x + 6 = 0$

Step 3.6.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 3.7

The final solution is all the values that make $x (x + 3) (x + 6) = 0$ true.

$x = 0, - 3, - 6$

Step 4

The values which make the derivative equal to $0$ are $0, - 3, - 6$ .

$0, - 3, - 6$

Step 5

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

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

Step 6

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

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Step 6.1

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

$f' (- 7) = {(- 7)}^{3} + 9 {(- 7)}^{2} + 18 (- 7)$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 7$ to the power of $3$ .

$f' (- 7) = - 343 + 9 {(- 7)}^{2} + 18 (- 7)$

Step 6.2.1.2

Raise $- 7$ to the power of $2$ .

$f' (- 7) = - 343 + 9 \cdot 49 + 18 (- 7)$

Step 6.2.1.3

Multiply $9$ by $49$ .

$f' (- 7) = - 343 + 441 + 18 (- 7)$

Step 6.2.1.4

Multiply $18$ by $- 7$ .

$f' (- 7) = - 343 + 441 - 126$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $- 343$ and $441$ .

$f' (- 7) = 98 - 126$

Step 6.2.2.2

Subtract $126$ from $98$ .

$f' (- 7) = - 28$

Step 6.2.3

The final answer is $- 28$ .

$- 28$

Step 6.3

At $x = - 7$ the derivative is $- 28$ . Since this is negative, the function is decreasing on $(- \infty, - 6)$ .

Decreasing on $(- \infty, - 6)$ since $f' (x) < 0$

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

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

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Step 7.2.1.1.1

Apply the product rule to $- \frac{9}{2}$ .

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

Step 7.2.1.1.2

Apply the product rule to $\frac{9}{2}$ .

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

Step 7.2.1.2

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

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

Step 7.2.1.3

Raise $9$ to the power of $3$ .

$f' (- \frac{9}{2}) = - \frac{729}{2^{3}} + 9 {(- \frac{9}{2})}^{2} + 18 (- \frac{9}{2})$

Step 7.2.1.4

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

$f' (- \frac{9}{2}) = - \frac{729}{8} + 9 {(- \frac{9}{2})}^{2} + 18 (- \frac{9}{2})$

Step 7.2.1.5

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

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Step 7.2.1.5.1

Apply the product rule to $- \frac{9}{2}$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + 9 ({(- 1)}^{2} {(\frac{9}{2})}^{2}) + 18 (- \frac{9}{2})$

Step 7.2.1.5.2

Apply the product rule to $\frac{9}{2}$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + 9 ({(- 1)}^{2} (\frac{9^{2}}{2^{2}})) + 18 (- \frac{9}{2})$

Step 7.2.1.6

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

$f' (- \frac{9}{2}) = - \frac{729}{8} + 9 (1 (\frac{9^{2}}{2^{2}})) + 18 (- \frac{9}{2})$

Step 7.2.1.7

Multiply $\frac{9^{2}}{2^{2}}$ by $1$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + 9 (\frac{9^{2}}{2^{2}}) + 18 (- \frac{9}{2})$

Step 7.2.1.8

Raise $9$ to the power of $2$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + 9 (\frac{81}{2^{2}}) + 18 (- \frac{9}{2})$

Step 7.2.1.9

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

$f' (- \frac{9}{2}) = - \frac{729}{8} + 9 (\frac{81}{4}) + 18 (- \frac{9}{2})$

Step 7.2.1.10

Multiply $9 (\frac{81}{4})$ .

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Step 7.2.1.10.1

Combine $9$ and $\frac{81}{4}$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{9 \cdot 81}{4} + 18 (- \frac{9}{2})$

Step 7.2.1.10.2

Multiply $9$ by $81$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729}{4} + 18 (- \frac{9}{2})$

Step 7.2.1.11

Cancel the common factor of $2$ .

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Step 7.2.1.11.1

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729}{4} + 18 (\frac{- 9}{2})$

Step 7.2.1.11.2

Factor $2$ out of $18$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729}{4} + 2 (9) (\frac{- 9}{2})$

Step 7.2.1.11.3

Cancel the common factor.

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729}{4} + 2 \cdot (9 (\frac{- 9}{2}))$

Step 7.2.1.11.4

Rewrite the expression.

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729}{4} + 9 \cdot - 9$

Step 7.2.1.12

Multiply $9$ by $- 9$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729}{4} - 81$

Step 7.2.2

Find the common denominator.

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Step 7.2.2.1

Multiply $\frac{729}{4}$ by $\frac{2}{2}$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729}{4} \cdot \frac{2}{2} - 81$

Step 7.2.2.2

Multiply $\frac{729}{4}$ by $\frac{2}{2}$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729 \cdot 2}{4 \cdot 2} - 81$

Step 7.2.2.3

Write $- 81$ as a fraction with denominator $1$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729 \cdot 2}{4 \cdot 2} + \frac{- 81}{1}$

Step 7.2.2.4

Multiply $\frac{- 81}{1}$ by $\frac{8}{8}$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729 \cdot 2}{4 \cdot 2} + \frac{- 81}{1} \cdot \frac{8}{8}$

Step 7.2.2.5

Multiply $\frac{- 81}{1}$ by $\frac{8}{8}$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729 \cdot 2}{4 \cdot 2} + \frac{- 81 \cdot 8}{8}$

Step 7.2.2.6

Reorder the factors of $4 \cdot 2$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729 \cdot 2}{2 \cdot 4} + \frac{- 81 \cdot 8}{8}$

Step 7.2.2.7

Multiply $2$ by $4$ .

$f' (- \frac{9}{2}) = - \frac{729}{8} + \frac{729 \cdot 2}{8} + \frac{- 81 \cdot 8}{8}$

Step 7.2.3

Combine the numerators over the common denominator.

$f' (- \frac{9}{2}) = \frac{- 729 + 729 \cdot 2 - 81 \cdot 8}{8}$

Step 7.2.4

Simplify each term.

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Step 7.2.4.1

Multiply $729$ by $2$ .

$f' (- \frac{9}{2}) = \frac{- 729 + 1458 - 81 \cdot 8}{8}$

Step 7.2.4.2

Multiply $- 81$ by $8$ .

$f' (- \frac{9}{2}) = \frac{- 729 + 1458 - 648}{8}$

Step 7.2.5

Simplify by adding and subtracting.

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Step 7.2.5.1

Add $- 729$ and $1458$ .

$f' (- \frac{9}{2}) = \frac{729 - 648}{8}$

Step 7.2.5.2

Subtract $648$ from $729$ .

$f' (- \frac{9}{2}) = \frac{81}{8}$

Step 7.2.6

The final answer is $\frac{81}{8}$ .

$\frac{81}{8}$

Step 7.3

At $x = - \frac{9}{2}$ the derivative is $\frac{81}{8}$ . Since this is positive, the function is increasing on $(- 6, - 3)$ .

Increasing on $(- 6, - 3)$ since $f' (x) > 0$

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

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

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Step 8.2.1.1.1

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

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

Step 8.2.1.1.2

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

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

Step 8.2.1.2

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

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

Step 8.2.1.3

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

$f' (- \frac{3}{2}) = - \frac{27}{2^{3}} + 9 {(- \frac{3}{2})}^{2} + 18 (- \frac{3}{2})$

Step 8.2.1.4

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

$f' (- \frac{3}{2}) = - \frac{27}{8} + 9 {(- \frac{3}{2})}^{2} + 18 (- \frac{3}{2})$

Step 8.2.1.5

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

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Step 8.2.1.5.1

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

$f' (- \frac{3}{2}) = - \frac{27}{8} + 9 ({(- 1)}^{2} {(\frac{3}{2})}^{2}) + 18 (- \frac{3}{2})$

Step 8.2.1.5.2

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

$f' (- \frac{3}{2}) = - \frac{27}{8} + 9 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}})) + 18 (- \frac{3}{2})$

Step 8.2.1.6

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

$f' (- \frac{3}{2}) = - \frac{27}{8} + 9 (1 (\frac{3^{2}}{2^{2}})) + 18 (- \frac{3}{2})$

Step 8.2.1.7

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$f' (- \frac{3}{2}) = - \frac{27}{8} + 9 (\frac{3^{2}}{2^{2}}) + 18 (- \frac{3}{2})$

Step 8.2.1.8

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

$f' (- \frac{3}{2}) = - \frac{27}{8} + 9 (\frac{9}{2^{2}}) + 18 (- \frac{3}{2})$

Step 8.2.1.9

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

$f' (- \frac{3}{2}) = - \frac{27}{8} + 9 (\frac{9}{4}) + 18 (- \frac{3}{2})$

Step 8.2.1.10

Multiply $9 (\frac{9}{4})$ .

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Step 8.2.1.10.1

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

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{9 \cdot 9}{4} + 18 (- \frac{3}{2})$

Step 8.2.1.10.2

Multiply $9$ by $9$ .

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81}{4} + 18 (- \frac{3}{2})$

Step 8.2.1.11

Cancel the common factor of $2$ .

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Step 8.2.1.11.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81}{4} + 18 (\frac{- 3}{2})$

Step 8.2.1.11.2

Factor $2$ out of $18$ .

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81}{4} + 2 (9) (\frac{- 3}{2})$

Step 8.2.1.11.3

Cancel the common factor.

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81}{4} + 2 \cdot (9 (\frac{- 3}{2}))$

Step 8.2.1.11.4

Rewrite the expression.

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81}{4} + 9 \cdot - 3$

Step 8.2.1.12

Multiply $9$ by $- 3$ .

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81}{4} - 27$

Step 8.2.2

Find the common denominator.

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Step 8.2.2.1

Multiply $\frac{81}{4}$ by $\frac{2}{2}$ .

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81}{4} \cdot \frac{2}{2} - 27$

Step 8.2.2.2

Multiply $\frac{81}{4}$ by $\frac{2}{2}$ .

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81 \cdot 2}{4 \cdot 2} - 27$

Step 8.2.2.3

Write $- 27$ as a fraction with denominator $1$ .

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81 \cdot 2}{4 \cdot 2} + \frac{- 27}{1}$

Step 8.2.2.4

Multiply $\frac{- 27}{1}$ by $\frac{8}{8}$ .

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81 \cdot 2}{4 \cdot 2} + \frac{- 27}{1} \cdot \frac{8}{8}$

Step 8.2.2.5

Multiply $\frac{- 27}{1}$ by $\frac{8}{8}$ .

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81 \cdot 2}{4 \cdot 2} + \frac{- 27 \cdot 8}{8}$

Step 8.2.2.6

Reorder the factors of $4 \cdot 2$ .

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81 \cdot 2}{2 \cdot 4} + \frac{- 27 \cdot 8}{8}$

Step 8.2.2.7

Multiply $2$ by $4$ .

$f' (- \frac{3}{2}) = - \frac{27}{8} + \frac{81 \cdot 2}{8} + \frac{- 27 \cdot 8}{8}$

Step 8.2.3

Combine the numerators over the common denominator.

$f' (- \frac{3}{2}) = \frac{- 27 + 81 \cdot 2 - 27 \cdot 8}{8}$

Step 8.2.4

Simplify each term.

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Step 8.2.4.1

Multiply $81$ by $2$ .

$f' (- \frac{3}{2}) = \frac{- 27 + 162 - 27 \cdot 8}{8}$

Step 8.2.4.2

Multiply $- 27$ by $8$ .

$f' (- \frac{3}{2}) = \frac{- 27 + 162 - 216}{8}$

Step 8.2.5

Simplify the expression.

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Step 8.2.5.1

Add $- 27$ and $162$ .

$f' (- \frac{3}{2}) = \frac{135 - 216}{8}$

Step 8.2.5.2

Subtract $216$ from $135$ .

$f' (- \frac{3}{2}) = \frac{- 81}{8}$

Step 8.2.5.3

Move the negative in front of the fraction.

$f' (- \frac{3}{2}) = - \frac{81}{8}$

Step 8.2.6

The final answer is $- \frac{81}{8}$ .

$- \frac{81}{8}$

Step 8.3

At $x = - \frac{3}{2}$ the derivative is $- \frac{81}{8}$ . Since this is negative, the function is decreasing on $(- 3, 0)$ .

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

Step 9

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

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Step 9.1

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

$f' (1) = {(1)}^{3} + 9 {(1)}^{2} + 18 (1)$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

One to any power is one.

$f' (1) = 1 + 9 {(1)}^{2} + 18 (1)$

Step 9.2.1.2

One to any power is one.

$f' (1) = 1 + 9 \cdot 1 + 18 (1)$

Step 9.2.1.3

Multiply $9$ by $1$ .

$f' (1) = 1 + 9 + 18 (1)$

Step 9.2.1.4

Multiply $18$ by $1$ .

$f' (1) = 1 + 9 + 18$

Step 9.2.2

Simplify by adding numbers.

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Step 9.2.2.1

Add $1$ and $9$ .

$f' (1) = 10 + 18$

Step 9.2.2.2

Add $10$ and $18$ .

$f' (1) = 28$

Step 9.2.3

The final answer is $28$ .

$28$

Step 9.3

At $x = 1$ the derivative is $28$ . Since this is positive, the function is increasing on $(0, \infty)$ .

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

Step 10

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

Increasing on: $(- 6, - 3), (0, \infty)$

Decreasing on: $(- \infty, - 6), (- 3, 0)$

Step 11