Calculus Examples

$P (x) = \ln (- x^{3} + 12 x^{2} + 27 x + 1)$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = - x^{3} + 12 x^{2} + 27 x + 1$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $- x^{3} + 12 x^{2} + 27 x + 1$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [- x^{3} + 12 x^{2} + 27 x + 1]$

Step 1.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [- x^{3} + 12 x^{2} + 27 x + 1]$

Step 1.1.3

Replace all occurrences of $u$ with $- x^{3} + 12 x^{2} + 27 x + 1$ .

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \frac{d}{d x} [- x^{3} + 12 x^{2} + 27 x + 1]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $- x^{3} + 12 x^{2} + 27 x + 1$ with respect to $x$ is $\frac{d}{d x} [- x^{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [27 x] + \frac{d}{d x} [1]$ .

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (\frac{d}{d x} [- x^{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [27 x] + \frac{d}{d x} [1])$

Step 1.2.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- \frac{d}{d x} [x^{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [27 x] + \frac{d}{d x} [1])$

Step 1.2.3

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

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- (3 x^{2}) + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [27 x] + \frac{d}{d x} [1])$

Step 1.2.4

Multiply $3$ by $- 1$ .

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- 3 x^{2} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [27 x] + \frac{d}{d x} [1])$

Step 1.2.5

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- 3 x^{2} + 12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [27 x] + \frac{d}{d x} [1])$

Step 1.2.6

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

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- 3 x^{2} + 12 (2 x) + \frac{d}{d x} [27 x] + \frac{d}{d x} [1])$

Step 1.2.7

Multiply $2$ by $12$ .

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- 3 x^{2} + 24 x + \frac{d}{d x} [27 x] + \frac{d}{d x} [1])$

Step 1.2.8

Since $27$ is constant with respect to $x$ , the derivative of $27 x$ with respect to $x$ is $27 \frac{d}{d x} [x]$ .

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- 3 x^{2} + 24 x + 27 \frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 1.2.9

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

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- 3 x^{2} + 24 x + 27 \cdot 1 + \frac{d}{d x} [1])$

Step 1.2.10

Multiply $27$ by $1$ .

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- 3 x^{2} + 24 x + 27 + \frac{d}{d x} [1])$

Step 1.2.11

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- 3 x^{2} + 24 x + 27 + 0)$

Step 1.2.12

Add $- 3 x^{2} + 24 x + 27$ and $0$ .

$\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- 3 x^{2} + 24 x + 27)$

Step 1.3

Simplify.

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Step 1.3.1

Reorder the factors of $\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- 3 x^{2} + 24 x + 27)$ .

$(- 3 x^{2} + 24 x + 27) \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.2

Multiply $- 3 x^{2} + 24 x + 27$ by $\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1}$ .

$\frac{- 3 x^{2} + 24 x + 27}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Factor $3$ out of $- 3 x^{2} + 24 x + 27$ .

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Step 1.3.3.1.1

Factor $3$ out of $- 3 x^{2}$ .

$\frac{3 (- x^{2}) + 24 x + 27}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.3.1.2

Factor $3$ out of $24 x$ .

$\frac{3 (- x^{2}) + 3 (8 x) + 27}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.3.1.3

Factor $3$ out of $27$ .

$\frac{3 (- x^{2}) + 3 (8 x) + 3 (9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.3.1.4

Factor $3$ out of $3 (- x^{2}) + 3 (8 x)$ .

$\frac{3 (- x^{2} + 8 x) + 3 (9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.3.1.5

Factor $3$ out of $3 (- x^{2} + 8 x) + 3 (9)$ .

$\frac{3 (- x^{2} + 8 x + 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.3.2

Factor by grouping.

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Step 1.3.3.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 9 = - 9$ and whose sum is $b = 8$ .

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Step 1.3.3.2.1.1

Factor $8$ out of $8 x$ .

$\frac{3 (- x^{2} + 8 (x) + 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.3.2.1.2

Rewrite $8$ as $- 1$ plus $9$

$\frac{3 (- x^{2} + (- 1 + 9) x + 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.3.2.1.3

Apply the distributive property.

$\frac{3 (- x^{2} - 1 x + 9 x + 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.3.2.2

Factor out the greatest common factor from each group.

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Step 1.3.3.2.2.1

Group the first two terms and the last two terms.

$\frac{3 ((- x^{2} - 1 x) + 9 x + 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.3.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{3 (x (- x - 1) - 9 (- x - 1))}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.3.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

$\frac{3 ((- x - 1) (x - 9))}{- x^{3} + 12 x^{2} + 27 x + 1}$

$\frac{3 (- x - 1) (x - 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.4

Factor $- 1$ out of $- x$ .

$\frac{3 (- (x) - 1) (x - 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.5

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{3 (- (x) - 1 (1)) (x - 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.6

Factor $- 1$ out of $- (x) - 1 (1)$ .

$\frac{3 (- (x + 1)) (x - 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.7

Rewrite $- (x + 1)$ as $- 1 (x + 1)$ .

$\frac{3 (- 1 (x + 1)) (x - 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 1.3.8

Factor $- 1$ out of $- x^{3}$ .

$\frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3}) + 12 x^{2} + 27 x + 1}$

Step 1.3.9

Factor $- 1$ out of $12 x^{2}$ .

$\frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3}) - (- 12 x^{2}) + 27 x + 1}$

Step 1.3.10

Factor $- 1$ out of $- (x^{3}) - (- 12 x^{2})$ .

$\frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3} - 12 x^{2}) + 27 x + 1}$

Step 1.3.11

Factor $- 1$ out of $27 x$ .

$\frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3} - 12 x^{2}) - (- 27 x) + 1}$

Step 1.3.12

Factor $- 1$ out of $- (x^{3} - 12 x^{2}) - (- 27 x)$ .

$\frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3} - 12 x^{2} - 27 x) + 1}$

Step 1.3.13

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3} - 12 x^{2} - 27 x) - 1 (- 1)}$

Step 1.3.14

Factor $- 1$ out of $- (x^{3} - 12 x^{2} - 27 x) - 1 (- 1)$ .

$\frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3} - 12 x^{2} - 27 x - 1)}$

Step 1.3.15

Rewrite $- (x^{3} - 12 x^{2} - 27 x - 1)$ as $- 1 (x^{3} - 12 x^{2} - 27 x - 1)$ .

$\frac{3 (- 1 (x + 1)) (x - 9)}{- 1 (x^{3} - 12 x^{2} - 27 x - 1)}$

Step 1.3.16

Cancel the common factor.

$\frac{3 (- 1 (x + 1)) (x - 9)}{- 1 (x^{3} - 12 x^{2} - 27 x - 1)}$

Step 1.3.17

Rewrite the expression.

$\frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{(x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1}]$ .

$P'' (x) = 3 \frac{d}{d x} (\frac{(x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1})$

Step 2.2

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 $f (x) = (x + 1) (x - 9)$ and $g (x) = x^{3} - 12 x^{2} - 27 x - 1$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) \frac{d}{d x} ((x + 1) (x - 9)) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 1$ and $g (x) = x - 9$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) ((x + 1) \frac{d}{d x} (x - 9) + (x - 9) \frac{d}{d x} (x + 1)) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4

Differentiate.

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Step 2.4.1

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

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) ((x + 1) (\frac{d}{d x} (x) + \frac{d}{d x} (- 9)) + (x - 9) \frac{d}{d x} (x + 1)) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.2

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

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) ((x + 1) (1 + \frac{d}{d x} (- 9)) + (x - 9) \frac{d}{d x} (x + 1)) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.3

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) ((x + 1) (1 + 0) + (x - 9) \frac{d}{d x} (x + 1)) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.4

Simplify the expression.

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Step 2.4.4.1

Add $1$ and $0$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) ((x + 1) \cdot 1 + (x - 9) \frac{d}{d x} (x + 1)) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.4.2

Multiply $x + 1$ by $1$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (x + 1 + (x - 9) \frac{d}{d x} (x + 1)) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.5

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (x + 1 + (x - 9) (\frac{d}{d x} (x) + \frac{d}{d x} (1))) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.6

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

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (x + 1 + (x - 9) (1 + \frac{d}{d x} (1))) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (x + 1 + (x - 9) (1 + 0)) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.8

Simplify by adding terms.

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Step 2.4.8.1

Add $1$ and $0$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (x + 1 + (x - 9) \cdot 1) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.8.2

Multiply $x - 9$ by $1$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (x + 1 + x - 9) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.8.3

Add $x$ and $x$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x + 1 - 9) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.8.4

Subtract $9$ from $1$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) \frac{d}{d x} (x^{3} - 12 x^{2} - 27 x - 1)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.9

By the Sum Rule, the derivative of $x^{3} - 12 x^{2} - 27 x - 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [- 27 x] + \frac{d}{d x} [- 1]$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) (\frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 12 x^{2}) + \frac{d}{d x} (- 27 x) + \frac{d}{d x} (- 1))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.10

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

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) (3 x^{2} + \frac{d}{d x} (- 12 x^{2}) + \frac{d}{d x} (- 27 x) + \frac{d}{d x} (- 1))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.11

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{2}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{2}]$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) (3 x^{2} - 12 \frac{d}{d x} x^{2} + \frac{d}{d x} (- 27 x) + \frac{d}{d x} (- 1))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.12

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

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) (3 x^{2} - 12 (2 x) + \frac{d}{d x} (- 27 x) + \frac{d}{d x} (- 1))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.13

Multiply $2$ by $- 12$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) (3 x^{2} - 24 x + \frac{d}{d x} (- 27 x) + \frac{d}{d x} (- 1))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.14

Since $- 27$ is constant with respect to $x$ , the derivative of $- 27 x$ with respect to $x$ is $- 27 \frac{d}{d x} [x]$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) (3 x^{2} - 24 x - 27 \frac{d}{d x} x + \frac{d}{d x} (- 1))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.15

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

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) (3 x^{2} - 24 x - 27 \cdot 1 + \frac{d}{d x} (- 1))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.16

Multiply $- 27$ by $1$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) (3 x^{2} - 24 x - 27 + \frac{d}{d x} (- 1))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.17

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) (3 x^{2} - 24 x - 27 + 0)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.18

Combine fractions.

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Step 2.4.18.1

Add $3 x^{2} - 24 x - 27$ and $0$ .

$P'' (x) = 3 (\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) (3 x^{2} - 24 x - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}})$

Step 2.4.18.2

Combine $3$ and $\frac{(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) (3 x^{2} - 24 x - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$ .

$P'' (x) = \frac{3 ((x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) - (x + 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$P'' (x) = \frac{3 ((x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8) + (- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.2

Apply the distributive property.

$P'' (x) = \frac{3 ((x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8)) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3

Simplify the numerator.

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Step 2.5.3.1

Simplify each term.

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Step 2.5.3.1.1

Expand $(x^{3} - 12 x^{2} - 27 x - 1) (2 x - 8)$ by multiplying each term in the first expression by each term in the second expression.

$P'' (x) = \frac{3 (x^{3} (2 x) + x^{3} \cdot - 8 - 12 x^{2} (2 x) - 12 x^{2} \cdot - 8 - 27 x (2 x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2

Simplify each term.

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Step 2.5.3.1.2.1

Rewrite using the commutative property of multiplication.

$P'' (x) = \frac{3 (2 x^{3} x + x^{3} \cdot - 8 - 12 x^{2} (2 x) - 12 x^{2} \cdot - 8 - 27 x (2 x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.2

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

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Step 2.5.3.1.2.2.1

Move $x$ .

$P'' (x) = \frac{3 (2 (x \cdot x^{3}) + x^{3} \cdot - 8 - 12 x^{2} (2 x) - 12 x^{2} \cdot - 8 - 27 x (2 x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.2.2

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

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Step 2.5.3.1.2.2.2.1

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

Step 2.5.3.1.2.2.2.2

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

$P'' (x) = \frac{3 (2 x^{1 + 3} + x^{3} \cdot - 8 - 12 x^{2} (2 x) - 12 x^{2} \cdot - 8 - 27 x (2 x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.2.3

Add $1$ and $3$ .

$P'' (x) = \frac{3 (2 x^{4} + x^{3} \cdot - 8 - 12 x^{2} (2 x) - 12 x^{2} \cdot - 8 - 27 x (2 x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.3

Move $- 8$ to the left of $x^{3}$ .

$P'' (x) = \frac{3 (2 x^{4} - 8 \cdot x^{3} - 12 x^{2} (2 x) - 12 x^{2} \cdot - 8 - 27 x (2 x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.4

Rewrite using the commutative property of multiplication.

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 12 \cdot (2 x^{2} x) - 12 x^{2} \cdot - 8 - 27 x (2 x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.5

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

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Step 2.5.3.1.2.5.1

Move $x$ .

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 12 \cdot (2 (x \cdot x^{2})) - 12 x^{2} \cdot - 8 - 27 x (2 x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.5.2

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

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Step 2.5.3.1.2.5.2.1

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

Step 2.5.3.1.2.5.2.2

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

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 12 \cdot (2 x^{1 + 2}) - 12 x^{2} \cdot - 8 - 27 x (2 x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.5.3

Add $1$ and $2$ .

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 12 \cdot (2 x^{3}) - 12 x^{2} \cdot - 8 - 27 x (2 x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.6

Multiply $- 12$ by $2$ .

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 24 x^{3} - 12 x^{2} \cdot - 8 - 27 x (2 x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.7

Multiply $- 8$ by $- 12$ .

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 24 x^{3} + 96 x^{2} - 27 x (2 x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.8

Rewrite using the commutative property of multiplication.

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 24 x^{3} + 96 x^{2} - 27 \cdot (2 x \cdot x) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.9

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

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Step 2.5.3.1.2.9.1

Move $x$ .

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 24 x^{3} + 96 x^{2} - 27 \cdot (2 (x \cdot x)) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.9.2

Multiply $x$ by $x$ .

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 24 x^{3} + 96 x^{2} - 27 \cdot (2 x^{2}) - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.10

Multiply $- 27$ by $2$ .

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 24 x^{3} + 96 x^{2} - 54 x^{2} - 27 x \cdot - 8 - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.11

Multiply $- 8$ by $- 27$ .

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 24 x^{3} + 96 x^{2} - 54 x^{2} + 216 x - 1 (2 x) - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.12

Multiply $2$ by $- 1$ .

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 24 x^{3} + 96 x^{2} - 54 x^{2} + 216 x - 2 x - 1 \cdot - 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.2.13

Multiply $- 1$ by $- 8$ .

$P'' (x) = \frac{3 (2 x^{4} - 8 x^{3} - 24 x^{3} + 96 x^{2} - 54 x^{2} + 216 x - 2 x + 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.3

Subtract $24 x^{3}$ from $- 8 x^{3}$ .

$P'' (x) = \frac{3 (2 x^{4} - 32 x^{3} + 96 x^{2} - 54 x^{2} + 216 x - 2 x + 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.4

Subtract $54 x^{2}$ from $96 x^{2}$ .

$P'' (x) = \frac{3 (2 x^{4} - 32 x^{3} + 42 x^{2} + 216 x - 2 x + 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.5

Subtract $2 x$ from $216 x$ .

$P'' (x) = \frac{3 (2 x^{4} - 32 x^{3} + 42 x^{2} + 214 x + 8) + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.6

Apply the distributive property.

$P'' (x) = \frac{3 (2 x^{4}) + 3 (- 32 x^{3}) + 3 (42 x^{2}) + 3 (214 x) + 3 \cdot 8 + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.7

Simplify.

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Step 2.5.3.1.7.1

Multiply $2$ by $3$ .

$P'' (x) = \frac{6 x^{4} + 3 (- 32 x^{3}) + 3 (42 x^{2}) + 3 (214 x) + 3 \cdot 8 + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.7.2

Multiply $- 32$ by $3$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 3 (42 x^{2}) + 3 (214 x) + 3 \cdot 8 + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.7.3

Multiply $42$ by $3$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 3 (214 x) + 3 \cdot 8 + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.7.4

Multiply $214$ by $3$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 3 \cdot 8 + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.7.5

Multiply $3$ by $8$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 ((- x - 1 \cdot 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.8

Multiply $- 1$ by $1$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 ((- x - 1) (x - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.9

Expand $(- x - 1) (x - 9)$ using the FOIL Method.

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Step 2.5.3.1.9.1

Apply the distributive property.

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 ((- x (x - 9) - 1 (x - 9)) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.9.2

Apply the distributive property.

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 ((- x \cdot x - x \cdot - 9 - 1 (x - 9)) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.9.3

Apply the distributive property.

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 ((- x \cdot x - x \cdot - 9 - 1 x - 1 \cdot - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.10

Simplify and combine like terms.

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Step 2.5.3.1.10.1

Simplify each term.

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Step 2.5.3.1.10.1.1

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

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Step 2.5.3.1.10.1.1.1

Move $x$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 ((- (x \cdot x) - x \cdot - 9 - 1 x - 1 \cdot - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.10.1.1.2

Multiply $x$ by $x$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 ((- x^{2} - x \cdot - 9 - 1 x - 1 \cdot - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.10.1.2

Multiply $- 9$ by $- 1$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 ((- x^{2} + 9 x - 1 x - 1 \cdot - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.10.1.3

Rewrite $- 1 x$ as $- x$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 ((- x^{2} + 9 x - x - 1 \cdot - 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.10.1.4

Multiply $- 1$ by $- 9$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 ((- x^{2} + 9 x - x + 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.10.2

Subtract $x$ from $9 x$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 ((- x^{2} + 8 x + 9) (3 x^{2} - 24 x - 27))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.11

Expand $(- x^{2} + 8 x + 9) (3 x^{2} - 24 x - 27)$ by multiplying each term in the first expression by each term in the second expression.

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- x^{2} (3 x^{2}) - x^{2} (- 24 x) - x^{2} \cdot - 27 + 8 x (3 x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12

Simplify each term.

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Step 2.5.3.1.12.1

Rewrite using the commutative property of multiplication.

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 1 \cdot (3 x^{2} x^{2}) - x^{2} (- 24 x) - x^{2} \cdot - 27 + 8 x (3 x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.2

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

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Step 2.5.3.1.12.2.1

Move $x^{2}$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 1 \cdot (3 (x^{2} x^{2})) - x^{2} (- 24 x) - x^{2} \cdot - 27 + 8 x (3 x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.2.2

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

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 1 \cdot (3 x^{2 + 2}) - x^{2} (- 24 x) - x^{2} \cdot - 27 + 8 x (3 x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.2.3

Add $2$ and $2$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 1 \cdot (3 x^{4}) - x^{2} (- 24 x) - x^{2} \cdot - 27 + 8 x (3 x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.3

Multiply $- 1$ by $3$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} - x^{2} (- 24 x) - x^{2} \cdot - 27 + 8 x (3 x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.4

Rewrite using the commutative property of multiplication.

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} - 1 \cdot (- 24 x^{2} x) - x^{2} \cdot - 27 + 8 x (3 x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.5

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

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Step 2.5.3.1.12.5.1

Move $x$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} - 1 \cdot (- 24 (x \cdot x^{2})) - x^{2} \cdot - 27 + 8 x (3 x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.5.2

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

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Step 2.5.3.1.12.5.2.1

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

Step 2.5.3.1.12.5.2.2

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

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} - 1 \cdot (- 24 x^{1 + 2}) - x^{2} \cdot - 27 + 8 x (3 x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.5.3

Add $1$ and $2$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} - 1 \cdot (- 24 x^{3}) - x^{2} \cdot - 27 + 8 x (3 x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.6

Multiply $- 1$ by $- 24$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} - x^{2} \cdot - 27 + 8 x (3 x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.7

Multiply $- 27$ by $- 1$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 8 x (3 x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.8

Rewrite using the commutative property of multiplication.

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 8 \cdot (3 x \cdot x^{2}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.9

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

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Step 2.5.3.1.12.9.1

Move $x^{2}$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 8 \cdot (3 (x^{2} x)) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.9.2

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

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Step 2.5.3.1.12.9.2.1

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

Step 2.5.3.1.12.9.2.2

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

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 8 \cdot (3 x^{2 + 1}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.9.3

Add $2$ and $1$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 8 \cdot (3 x^{3}) + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.10

Multiply $8$ by $3$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 24 x^{3} + 8 x (- 24 x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.11

Rewrite using the commutative property of multiplication.

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 24 x^{3} + 8 \cdot (- 24 x \cdot x) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.12

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

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Step 2.5.3.1.12.12.1

Move $x$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 24 x^{3} + 8 \cdot (- 24 (x \cdot x)) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.12.2

Multiply $x$ by $x$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 24 x^{3} + 8 \cdot (- 24 x^{2}) + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.13

Multiply $8$ by $- 24$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 24 x^{3} - 192 x^{2} + 8 x \cdot - 27 + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.14

Multiply $- 27$ by $8$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 24 x^{3} - 192 x^{2} - 216 x + 9 (3 x^{2}) + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.15

Multiply $3$ by $9$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 24 x^{3} - 192 x^{2} - 216 x + 27 x^{2} + 9 (- 24 x) + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.16

Multiply $- 24$ by $9$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 24 x^{3} - 192 x^{2} - 216 x + 27 x^{2} - 216 x + 9 \cdot - 27)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.12.17

Multiply $9$ by $- 27$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 24 x^{3} + 27 x^{2} + 24 x^{3} - 192 x^{2} - 216 x + 27 x^{2} - 216 x - 243)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.13

Add $24 x^{3}$ and $24 x^{3}$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 48 x^{3} + 27 x^{2} - 192 x^{2} - 216 x + 27 x^{2} - 216 x - 243)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.14

Subtract $192 x^{2}$ from $27 x^{2}$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 48 x^{3} - 165 x^{2} - 216 x + 27 x^{2} - 216 x - 243)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.15

Add $- 165 x^{2}$ and $27 x^{2}$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 48 x^{3} - 138 x^{2} - 216 x - 216 x - 243)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.16

Subtract $216 x$ from $- 216 x$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4} + 48 x^{3} - 138 x^{2} - 432 x - 243)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.17

Apply the distributive property.

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 3 (- 3 x^{4}) + 3 (48 x^{3}) + 3 (- 138 x^{2}) + 3 (- 432 x) + 3 \cdot - 243}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.18

Simplify.

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Step 2.5.3.1.18.1

Multiply $- 3$ by $3$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 - 9 x^{4} + 3 (48 x^{3}) + 3 (- 138 x^{2}) + 3 (- 432 x) + 3 \cdot - 243}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.18.2

Multiply $48$ by $3$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 - 9 x^{4} + 144 x^{3} + 3 (- 138 x^{2}) + 3 (- 432 x) + 3 \cdot - 243}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.18.3

Multiply $- 138$ by $3$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 - 9 x^{4} + 144 x^{3} - 414 x^{2} + 3 (- 432 x) + 3 \cdot - 243}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.18.4

Multiply $- 432$ by $3$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 - 9 x^{4} + 144 x^{3} - 414 x^{2} - 1296 x + 3 \cdot - 243}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.1.18.5

Multiply $3$ by $- 243$ .

$P'' (x) = \frac{6 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 - 9 x^{4} + 144 x^{3} - 414 x^{2} - 1296 x - 729}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.2

Subtract $9 x^{4}$ from $6 x^{4}$ .

$P'' (x) = \frac{- 3 x^{4} - 96 x^{3} + 126 x^{2} + 642 x + 24 + 144 x^{3} - 414 x^{2} - 1296 x - 729}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.3

Add $- 96 x^{3}$ and $144 x^{3}$ .

$P'' (x) = \frac{- 3 x^{4} + 48 x^{3} + 126 x^{2} + 642 x + 24 - 414 x^{2} - 1296 x - 729}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.4

Subtract $414 x^{2}$ from $126 x^{2}$ .

$P'' (x) = \frac{- 3 x^{4} + 48 x^{3} - 288 x^{2} + 642 x + 24 - 1296 x - 729}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.5

Subtract $1296 x$ from $642 x$ .

$P'' (x) = \frac{- 3 x^{4} + 48 x^{3} - 288 x^{2} - 654 x + 24 - 729}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.3.6

Subtract $729$ from $24$ .

$P'' (x) = \frac{- 3 x^{4} + 48 x^{3} - 288 x^{2} - 654 x - 705}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.4

Factor $3$ out of $- 3 x^{4} + 48 x^{3} - 288 x^{2} - 654 x - 705$ .

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Step 2.5.4.1

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

$P'' (x) = \frac{3 (- x^{4}) + 48 x^{3} - 288 x^{2} - 654 x - 705}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.4.2

Factor $3$ out of $48 x^{3}$ .

$P'' (x) = \frac{3 (- x^{4}) + 3 (16 x^{3}) - 288 x^{2} - 654 x - 705}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.4.3

Factor $3$ out of $- 288 x^{2}$ .

$P'' (x) = \frac{3 (- x^{4}) + 3 (16 x^{3}) + 3 (- 96 x^{2}) - 654 x - 705}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.4.4

Factor $3$ out of $- 654 x$ .

$P'' (x) = \frac{3 (- x^{4}) + 3 (16 x^{3}) + 3 (- 96 x^{2}) + 3 (- 218 x) - 705}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.4.5

Factor $3$ out of $- 705$ .

$P'' (x) = \frac{3 (- x^{4}) + 3 (16 x^{3}) + 3 (- 96 x^{2}) + 3 (- 218 x) + 3 (- 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.4.6

Factor $3$ out of $3 (- x^{4}) + 3 (16 x^{3})$ .

$P'' (x) = \frac{3 (- x^{4} + 16 x^{3}) + 3 (- 96 x^{2}) + 3 (- 218 x) + 3 (- 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.4.7

Factor $3$ out of $3 (- x^{4} + 16 x^{3}) + 3 (- 96 x^{2})$ .

$P'' (x) = \frac{3 (- x^{4} + 16 x^{3} - 96 x^{2}) + 3 (- 218 x) + 3 (- 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.4.8

Factor $3$ out of $3 (- x^{4} + 16 x^{3} - 96 x^{2}) + 3 (- 218 x)$ .

$P'' (x) = \frac{3 (- x^{4} + 16 x^{3} - 96 x^{2} - 218 x) + 3 (- 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.4.9

Factor $3$ out of $3 (- x^{4} + 16 x^{3} - 96 x^{2} - 218 x) + 3 (- 235)$ .

$P'' (x) = \frac{3 (- x^{4} + 16 x^{3} - 96 x^{2} - 218 x - 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.5

Factor $- 1$ out of $- x^{4}$ .

$P'' (x) = \frac{3 (- (x^{4}) + 16 x^{3} - 96 x^{2} - 218 x - 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.6

Factor $- 1$ out of $16 x^{3}$ .

$P'' (x) = \frac{3 (- (x^{4}) - (- 16 x^{3}) - 96 x^{2} - 218 x - 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.7

Factor $- 1$ out of $- (x^{4}) - (- 16 x^{3})$ .

$P'' (x) = \frac{3 (- (x^{4} - 16 x^{3}) - 96 x^{2} - 218 x - 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.8

Factor $- 1$ out of $- 96 x^{2}$ .

$P'' (x) = \frac{3 (- (x^{4} - 16 x^{3}) - (96 x^{2}) - 218 x - 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.9

Factor $- 1$ out of $- (x^{4} - 16 x^{3}) - (96 x^{2})$ .

$P'' (x) = \frac{3 (- (x^{4} - 16 x^{3} + 96 x^{2}) - 218 x - 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.10

Factor $- 1$ out of $- 218 x$ .

$P'' (x) = \frac{3 (- (x^{4} - 16 x^{3} + 96 x^{2}) - (218 x) - 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.11

Factor $- 1$ out of $- (x^{4} - 16 x^{3} + 96 x^{2}) - (218 x)$ .

$P'' (x) = \frac{3 (- (x^{4} - 16 x^{3} + 96 x^{2} + 218 x) - 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.12

Rewrite $- 235$ as $- 1 (235)$ .

$P'' (x) = \frac{3 (- (x^{4} - 16 x^{3} + 96 x^{2} + 218 x) - 1 \cdot 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.13

Factor $- 1$ out of $- (x^{4} - 16 x^{3} + 96 x^{2} + 218 x) - 1 (235)$ .

$P'' (x) = \frac{3 (- (x^{4} - 16 x^{3} + 96 x^{2} + 218 x + 235))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.14

Rewrite $- (x^{4} - 16 x^{3} + 96 x^{2} + 218 x + 235)$ as $- 1 (x^{4} - 16 x^{3} + 96 x^{2} + 218 x + 235)$ .

$P'' (x) = \frac{3 (- 1 (x^{4} - 16 x^{3} + 96 x^{2} + 218 x + 235))}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 2.5.15

Move the negative in front of the fraction.

$P'' (x) = - \frac{3 (x^{4} - 16 x^{3} + 96 x^{2} + 218 x + 235)}{{(x^{3} - 12 x^{2} - 27 x - 1)}^{2}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = - x^{3} + 12 x^{2} + 27 x + 1$ .

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Step 4.1.1.1

To apply the Chain Rule, set $u$ as $- x^{3} + 12 x^{2} + 27 x + 1$ .

$f' (x) = \frac{d}{d u} (\ln (u)) \frac{d}{d x} (- x^{3} + 12 x^{2} + 27 x + 1)$

Step 4.1.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$f' (x) = \frac{1}{u} \cdot \frac{d}{d x} (- x^{3} + 12 x^{2} + 27 x + 1)$

Step 4.1.1.3

Replace all occurrences of $u$ with $- x^{3} + 12 x^{2} + 27 x + 1$ .

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot \frac{d}{d x} (- x^{3} + 12 x^{2} + 27 x + 1)$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of $- x^{3} + 12 x^{2} + 27 x + 1$ with respect to $x$ is $\frac{d}{d x} [- x^{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [27 x] + \frac{d}{d x} [1]$ .

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot (\frac{d}{d x} (- x^{3}) + \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (27 x) + \frac{d}{d x} (1))$

Step 4.1.2.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot (- \frac{d}{d x} x^{3} + \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (27 x) + \frac{d}{d x} (1))$

Step 4.1.2.3

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

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot (- (3 x^{2}) + \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (27 x) + \frac{d}{d x} (1))$

Step 4.1.2.4

Multiply $3$ by $- 1$ .

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot (- 3 x^{2} + \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (27 x) + \frac{d}{d x} (1))$

Step 4.1.2.5

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot (- 3 x^{2} + 12 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (27 x) + \frac{d}{d x} (1))$

Step 4.1.2.6

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

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot (- 3 x^{2} + 12 (2 x) + \frac{d}{d x} (27 x) + \frac{d}{d x} (1))$

Step 4.1.2.7

Multiply $2$ by $12$ .

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot (- 3 x^{2} + 24 x + \frac{d}{d x} (27 x) + \frac{d}{d x} (1))$

Step 4.1.2.8

Since $27$ is constant with respect to $x$ , the derivative of $27 x$ with respect to $x$ is $27 \frac{d}{d x} [x]$ .

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot (- 3 x^{2} + 24 x + 27 \frac{d}{d x} (x) + \frac{d}{d x} (1))$

Step 4.1.2.9

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

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot (- 3 x^{2} + 24 x + 27 \cdot 1 + \frac{d}{d x} (1))$

Step 4.1.2.10

Multiply $27$ by $1$ .

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot (- 3 x^{2} + 24 x + 27 + \frac{d}{d x} (1))$

Step 4.1.2.11

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot (- 3 x^{2} + 24 x + 27 + 0)$

Step 4.1.2.12

Add $- 3 x^{2} + 24 x + 27$ and $0$ .

$f' (x) = \frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} \cdot (- 3 x^{2} + 24 x + 27)$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Reorder the factors of $\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1} (- 3 x^{2} + 24 x + 27)$ .

$f' (x) = (- 3 x^{2} + 24 x + 27) (\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1})$

Step 4.1.3.2

Multiply $- 3 x^{2} + 24 x + 27$ by $\frac{1}{- x^{3} + 12 x^{2} + 27 x + 1}$ .

$f' (x) = \frac{- 3 x^{2} + 24 x + 27}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.3

Simplify the numerator.

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Step 4.1.3.3.1

Factor $3$ out of $- 3 x^{2} + 24 x + 27$ .

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Step 4.1.3.3.1.1

Factor $3$ out of $- 3 x^{2}$ .

$f' (x) = \frac{3 (- x^{2}) + 24 x + 27}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.3.1.2

Factor $3$ out of $24 x$ .

$f' (x) = \frac{3 (- x^{2}) + 3 (8 x) + 27}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.3.1.3

Factor $3$ out of $27$ .

$f' (x) = \frac{3 (- x^{2}) + 3 (8 x) + 3 (9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.3.1.4

Factor $3$ out of $3 (- x^{2}) + 3 (8 x)$ .

$f' (x) = \frac{3 (- x^{2} + 8 x) + 3 (9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.3.1.5

Factor $3$ out of $3 (- x^{2} + 8 x) + 3 (9)$ .

$f' (x) = \frac{3 (- x^{2} + 8 x + 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.3.2

Factor by grouping.

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Step 4.1.3.3.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 9 = - 9$ and whose sum is $b = 8$ .

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Step 4.1.3.3.2.1.1

Factor $8$ out of $8 x$ .

$f' (x) = \frac{3 (- x^{2} + 8 (x) + 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.3.2.1.2

Rewrite $8$ as $- 1$ plus $9$

$f' (x) = \frac{3 (- x^{2} + (- 1 + 9) x + 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.3.2.1.3

Apply the distributive property.

$f' (x) = \frac{3 (- x^{2} - 1 x + 9 x + 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.3.2.2

Factor out the greatest common factor from each group.

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Step 4.1.3.3.2.2.1

Group the first two terms and the last two terms.

$f' (x) = \frac{3 ((- x^{2} - 1 x) + 9 x + 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.3.2.2.2

Factor out the greatest common factor (GCF) from each group.

$f' (x) = \frac{3 (x (- x - 1) - 9 (- x - 1))}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.3.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

$f' (x) = \frac{3 ((- x - 1) (x - 9))}{- x^{3} + 12 x^{2} + 27 x + 1}$

$f' (x) = \frac{3 (- x - 1) (x - 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.4

Factor $- 1$ out of $- x$ .

$f' (x) = \frac{3 (- (x) - 1) (x - 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.5

Rewrite $- 1$ as $- 1 (1)$ .

$f' (x) = \frac{3 (- (x) - 1 \cdot 1) (x - 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.6

Factor $- 1$ out of $- (x) - 1 (1)$ .

$f' (x) = \frac{3 (- (x + 1)) (x - 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.7

Rewrite $- (x + 1)$ as $- 1 (x + 1)$ .

$f' (x) = \frac{3 (- 1 (x + 1)) (x - 9)}{- x^{3} + 12 x^{2} + 27 x + 1}$

Step 4.1.3.8

Factor $- 1$ out of $- x^{3}$ .

$f' (x) = \frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3}) + 12 x^{2} + 27 x + 1}$

Step 4.1.3.9

Factor $- 1$ out of $12 x^{2}$ .

$f' (x) = \frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3}) - (- 12 x^{2}) + 27 x + 1}$

Step 4.1.3.10

Factor $- 1$ out of $- (x^{3}) - (- 12 x^{2})$ .

$f' (x) = \frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3} - 12 x^{2}) + 27 x + 1}$

Step 4.1.3.11

Factor $- 1$ out of $27 x$ .

$f' (x) = \frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3} - 12 x^{2}) - (- 27 x) + 1}$

Step 4.1.3.12

Factor $- 1$ out of $- (x^{3} - 12 x^{2}) - (- 27 x)$ .

$f' (x) = \frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3} - 12 x^{2} - 27 x) + 1}$

Step 4.1.3.13

Rewrite $1$ as $- 1 (- 1)$ .

$f' (x) = \frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3} - 12 x^{2} - 27 x) - 1 \cdot - 1}$

Step 4.1.3.14

Factor $- 1$ out of $- (x^{3} - 12 x^{2} - 27 x) - 1 (- 1)$ .

$f' (x) = \frac{3 (- 1 (x + 1)) (x - 9)}{- (x^{3} - 12 x^{2} - 27 x - 1)}$

Step 4.1.3.15

Rewrite $- (x^{3} - 12 x^{2} - 27 x - 1)$ as $- 1 (x^{3} - 12 x^{2} - 27 x - 1)$ .

$f' (x) = \frac{3 (- 1 (x + 1)) (x - 9)}{- 1 (x^{3} - 12 x^{2} - 27 x - 1)}$

Step 4.1.3.16

Cancel the common factor.

$f' (x) = \frac{3 (- 1 (x + 1)) (x - 9)}{- 1 (x^{3} - 12 x^{2} - 27 x - 1)}$

Step 4.1.3.17

Rewrite the expression.

$f' (x) = \frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1}$

Step 4.2

The first derivative of $P (x)$ with respect to $x$ is $\frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1}$ .

$\frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1} = 0$

Step 5.2

Set the numerator equal to zero.

$3 (x + 1) (x - 9) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

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

$x + 1 = 0$

$x - 9 = 0$

Step 5.3.2

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 5.3.2.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 5.3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5.3.3

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 5.3.3.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 5.3.3.2

Add $9$ to both sides of the equation.

$x = 9$

Step 5.3.4

The final solution is all the values that make $3 (x + 1) (x - 9) = 0$ true.

$x = - 1, 9$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1}$ equal to $0$ to find where the expression is undefined.

$x^{3} - 12 x^{2} - 27 x - 1 = 0$

Step 6.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 1.90410133, - 0.03766968, 13.94177101$

Step 6.3

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 $0$ .

$x = - 1.90410133, x = - 0.03766968, x = 13.94177101$

Step 7

Critical points to evaluate.

$x = 9, - 1.90410133, - 0.03766968$

Step 8

Evaluate the second derivative at $x = 9$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{3 ({(9)}^{4} - 16 {(9)}^{3} + 96 {(9)}^{2} + 218 (9) + 235)}{{({(9)}^{3} - 12 {(9)}^{2} - 27 \cdot 9 - 1)}^{2}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $9$ to the power of $4$ .

$- \frac{3 (6561 - 16 \cdot 9^{3} + 96 \cdot 9^{2} + 218 (9) + 235)}{{(9^{3} - 12 \cdot 9^{2} - 27 \cdot 9 - 1)}^{2}}$

Step 9.1.2

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

$- \frac{3 (6561 - 16 \cdot 729 + 96 \cdot 9^{2} + 218 (9) + 235)}{{(9^{3} - 12 \cdot 9^{2} - 27 \cdot 9 - 1)}^{2}}$

Step 9.1.3

Multiply $- 16$ by $729$ .

$- \frac{3 (6561 - 11664 + 96 \cdot 9^{2} + 218 (9) + 235)}{{(9^{3} - 12 \cdot 9^{2} - 27 \cdot 9 - 1)}^{2}}$

Step 9.1.4

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

$- \frac{3 (6561 - 11664 + 96 \cdot 81 + 218 (9) + 235)}{{(9^{3} - 12 \cdot 9^{2} - 27 \cdot 9 - 1)}^{2}}$

Step 9.1.5

Multiply $96$ by $81$ .

$- \frac{3 (6561 - 11664 + 7776 + 218 (9) + 235)}{{(9^{3} - 12 \cdot 9^{2} - 27 \cdot 9 - 1)}^{2}}$

Step 9.1.6

Multiply $218$ by $9$ .

$- \frac{3 (6561 - 11664 + 7776 + 1962 + 235)}{{(9^{3} - 12 \cdot 9^{2} - 27 \cdot 9 - 1)}^{2}}$

Step 9.1.7

Subtract $11664$ from $6561$ .

$- \frac{3 (- 5103 + 7776 + 1962 + 235)}{{(9^{3} - 12 \cdot 9^{2} - 27 \cdot 9 - 1)}^{2}}$

Step 9.1.8

Add $- 5103$ and $7776$ .

$- \frac{3 (2673 + 1962 + 235)}{{(9^{3} - 12 \cdot 9^{2} - 27 \cdot 9 - 1)}^{2}}$

Step 9.1.9

Add $2673$ and $1962$ .

$- \frac{3 (4635 + 235)}{{(9^{3} - 12 \cdot 9^{2} - 27 \cdot 9 - 1)}^{2}}$

Step 9.1.10

Add $4635$ and $235$ .

$- \frac{3 \cdot 4870}{{(9^{3} - 12 \cdot 9^{2} - 27 \cdot 9 - 1)}^{2}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

$- \frac{3 \cdot 4870}{{(729 - 12 \cdot 9^{2} - 27 \cdot 9 - 1)}^{2}}$

Step 9.2.2

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

$- \frac{3 \cdot 4870}{{(729 - 12 \cdot 81 - 27 \cdot 9 - 1)}^{2}}$

Step 9.2.3

Multiply $- 12$ by $81$ .

$- \frac{3 \cdot 4870}{{(729 - 972 - 27 \cdot 9 - 1)}^{2}}$

Step 9.2.4

Multiply $- 27$ by $9$ .

$- \frac{3 \cdot 4870}{{(729 - 972 - 243 - 1)}^{2}}$

Step 9.2.5

Subtract $972$ from $729$ .

$- \frac{3 \cdot 4870}{{(- 243 - 243 - 1)}^{2}}$

Step 9.2.6

Subtract $243$ from $- 243$ .

$- \frac{3 \cdot 4870}{{(- 486 - 1)}^{2}}$

Step 9.2.7

Subtract $1$ from $- 486$ .

$- \frac{3 \cdot 4870}{{(- 487)}^{2}}$

Step 9.2.8

Raise $- 487$ to the power of $2$ .

$- \frac{3 \cdot 4870}{237169}$

Step 9.3

Reduce the expression by cancelling the common factors.

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Step 9.3.1

Multiply $3$ by $4870$ .

$- \frac{14610}{237169}$

Step 9.3.2

Cancel the common factor of $14610$ and $237169$ .

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Step 9.3.2.1

Factor $487$ out of $14610$ .

$- \frac{487 (30)}{237169}$

Step 9.3.2.2

Cancel the common factors.

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Step 9.3.2.2.1

Factor $487$ out of $237169$ .

$- \frac{487 \cdot 30}{487 \cdot 487}$

Step 9.3.2.2.2

Cancel the common factor.

$- \frac{487 \cdot 30}{487 \cdot 487}$

Step 9.3.2.2.3

Rewrite the expression.

$- \frac{30}{487}$

Step 10

$x = 9$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 9$ is a local maximum

Step 11

Find the y-value when $x = 9$ .

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Step 11.1

Replace the variable $x$ with $9$ in the expression.

$P (9) = \ln (- {(9)}^{3} + 12 {(9)}^{2} + 27 (9) + 1)$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

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

$P (9) = \ln (- 1 \cdot 729 + 12 {(9)}^{2} + 27 (9) + 1)$

Step 11.2.1.2

Multiply $- 1$ by $729$ .

$P (9) = \ln (- 729 + 12 {(9)}^{2} + 27 (9) + 1)$

Step 11.2.1.3

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

$P (9) = \ln (- 729 + 12 \cdot 81 + 27 (9) + 1)$

Step 11.2.1.4

Multiply $12$ by $81$ .

$P (9) = \ln (- 729 + 972 + 27 (9) + 1)$

Step 11.2.1.5

Multiply $27$ by $9$ .

$P (9) = \ln (- 729 + 972 + 243 + 1)$

Step 11.2.2

Simplify by adding numbers.

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Step 11.2.2.1

Add $- 729$ and $972$ .

$P (9) = \ln (243 + 243 + 1)$

Step 11.2.2.2

Add $243$ and $243$ .

$P (9) = \ln (486 + 1)$

Step 11.2.2.3

Add $486$ and $1$ .

$P (9) = \ln (487)$

Step 11.2.3

The final answer is $\ln (487)$ .

$y = \ln (487)$

Step 12

Evaluate the second derivative at $x = - 1.90410133$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{3 ({(- 1.90410133)}^{4} - 16 {(- 1.90410133)}^{3} + 96 {(- 1.90410133)}^{2} + 218 (- 1.90410133) + 235)}{{({(- 1.90410133)}^{3} - 12 {(- 1.90410133)}^{2} - 27 \cdot - 1.90410133 - 1)}^{2}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Raise $- 1.90410133$ to the power of $3$ .

$- \frac{3 ({(- 1.90410133)}^{4} - 16 {(- 1.90410133)}^{3} + 96 {(- 1.90410133)}^{2} + 218 (- 1.90410133) + 235)}{{(- 6.90351337 - 12 {(- 1.90410133)}^{2} - 27 \cdot - 1.90410133 - 1)}^{2}}$

Step 13.1.2

Raise $- 1.90410133$ to the power of $2$ .

$- \frac{3 ({(- 1.90410133)}^{4} - 16 {(- 1.90410133)}^{3} + 96 {(- 1.90410133)}^{2} + 218 (- 1.90410133) + 235)}{{(- 6.90351337 - 12 \cdot 3.62560188 - 27 \cdot - 1.90410133 - 1)}^{2}}$

Step 13.1.3

Multiply $- 12$ by $3.62560188$ .

$- \frac{3 ({(- 1.90410133)}^{4} - 16 {(- 1.90410133)}^{3} + 96 {(- 1.90410133)}^{2} + 218 (- 1.90410133) + 235)}{{(- 6.90351337 - 43.50722259 - 27 \cdot - 1.90410133 - 1)}^{2}}$

Step 13.1.4

Multiply $- 27$ by $- 1.90410133$ .

$- \frac{3 ({(- 1.90410133)}^{4} - 16 {(- 1.90410133)}^{3} + 96 {(- 1.90410133)}^{2} + 218 (- 1.90410133) + 235)}{{(- 6.90351337 - 43.50722259 + 51.41073596 - 1)}^{2}}$

Step 13.2

Simplify the expression.

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Step 13.2.1

Subtract $43.50722259$ from $- 6.90351337$ .

$- \frac{3 ({(- 1.90410133)}^{4} - 16 {(- 1.90410133)}^{3} + 96 {(- 1.90410133)}^{2} + 218 (- 1.90410133) + 235)}{{(- 50.41073596 + 51.41073596 - 1)}^{2}}$

Step 13.2.2

Add $- 50.41073596$ and $51.41073596$ .

$- \frac{3 ({(- 1.90410133)}^{4} - 16 {(- 1.90410133)}^{3} + 96 {(- 1.90410133)}^{2} + 218 (- 1.90410133) + 235)}{{(0.99999999 - 1)}^{2}}$

Step 13.2.3

Subtract $1$ from $0.99999999$ .

$- \frac{3 ({(- 1.90410133)}^{4} - 16 {(- 1.90410133)}^{3} + 96 {(- 1.90410133)}^{2} + 218 (- 1.90410133) + 235)}{{(0)}^{2}}$

Step 13.2.4

Raise $0$ to the power of $2$ .

$- \frac{3 ({(- 1.90410133)}^{4} - 16 {(- 1.90410133)}^{3} + 96 {(- 1.90410133)}^{2} + 218 (- 1.90410133) + 235)}{0}$

Step 13.2.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{3 ({(- 1.90410133)}^{4} - 16 {(- 1.90410133)}^{3} + 96 {(- 1.90410133)}^{2} + 218 (- 1.90410133) + 235)}{0}$

Step 13.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 14

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 14.1

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

$(- \infty, - 1.90410133) \cup (- 1.90410133, - 0.03766968) \cup (- 0.03766968, 9) \cup (9, \infty)$

Step 14.2

Substitute any number, such as $- 4$ , from the interval $(- \infty, - 1.90410133)$ in the first derivative $\frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1}$ to check if the result is negative or positive.

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Step 14.2.1

Replace the variable $x$ with $- 4$ in the expression.

$P' (- 4) = \frac{3 ((- 4) + 1) ((- 4) - 9)}{{(- 4)}^{3} - 12 {(- 4)}^{2} - 27 \cdot - 4 - 1}$

Step 14.2.2

Simplify the result.

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Step 14.2.2.1

Simplify the numerator.

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Step 14.2.2.1.1

Add $- 4$ and $1$ .

$P' (- 4) = \frac{3 \cdot (- 3 (- 4 - 9))}{{(- 4)}^{3} - 12 {(- 4)}^{2} - 27 \cdot - 4 - 1}$

Step 14.2.2.1.2

Multiply $3$ by $- 3$ .

$P' (- 4) = \frac{- 9 (- 4 - 9)}{{(- 4)}^{3} - 12 {(- 4)}^{2} - 27 \cdot - 4 - 1}$

Step 14.2.2.1.3

Subtract $9$ from $- 4$ .

$P' (- 4) = \frac{- 9 \cdot - 13}{{(- 4)}^{3} - 12 {(- 4)}^{2} - 27 \cdot - 4 - 1}$

Step 14.2.2.2

Simplify the denominator.

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Step 14.2.2.2.1

Raise $- 4$ to the power of $3$ .

$P' (- 4) = \frac{- 9 \cdot - 13}{- 64 - 12 {(- 4)}^{2} - 27 \cdot - 4 - 1}$

Step 14.2.2.2.2

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

$P' (- 4) = \frac{- 9 \cdot - 13}{- 64 - 12 \cdot 16 - 27 \cdot - 4 - 1}$

Step 14.2.2.2.3

Multiply $- 12$ by $16$ .

$P' (- 4) = \frac{- 9 \cdot - 13}{- 64 - 192 - 27 \cdot - 4 - 1}$

Step 14.2.2.2.4

Multiply $- 27$ by $- 4$ .

$P' (- 4) = \frac{- 9 \cdot - 13}{- 64 - 192 + 108 - 1}$

Step 14.2.2.2.5

Subtract $192$ from $- 64$ .

$P' (- 4) = \frac{- 9 \cdot - 13}{- 256 + 108 - 1}$

Step 14.2.2.2.6

Add $- 256$ and $108$ .

$P' (- 4) = \frac{- 9 \cdot - 13}{- 148 - 1}$

Step 14.2.2.2.7

Subtract $1$ from $- 148$ .

$P' (- 4) = \frac{- 9 \cdot - 13}{- 149}$

Step 14.2.2.3

Simplify the expression.

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Step 14.2.2.3.1

Multiply $- 9$ by $- 13$ .

$P' (- 4) = \frac{117}{- 149}$

Step 14.2.2.3.2

Move the negative in front of the fraction.

$P' (- 4) = - \frac{117}{149}$

Step 14.2.2.4

The final answer is $- \frac{117}{149}$ .

$- \frac{117}{149}$

Step 14.3

Substitute any number, such as $- 1$ , from the interval $(- 1.90410133, - 0.03766968)$ in the first derivative $\frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1}$ to check if the result is negative or positive.

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Step 14.3.1

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

$P' (- 1) = \frac{3 ((- 1) + 1) ((- 1) - 9)}{{(- 1)}^{3} - 12 {(- 1)}^{2} - 27 \cdot - 1 - 1}$

Step 14.3.2

Simplify the result.

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Step 14.3.2.1

Simplify the numerator.

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Step 14.3.2.1.1

Add $- 1$ and $1$ .

$P' (- 1) = \frac{3 \cdot (0 (- 1 - 9))}{{(- 1)}^{3} - 12 {(- 1)}^{2} - 27 \cdot - 1 - 1}$

Step 14.3.2.1.2

Combine exponents.

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Step 14.3.2.1.2.1

Multiply $3$ by $0$ .

$P' (- 1) = \frac{0 (- 1 - 9)}{{(- 1)}^{3} - 12 {(- 1)}^{2} - 27 \cdot - 1 - 1}$

Step 14.3.2.1.2.2

Multiply $0$ by $- 1 - 9$ .

$P' (- 1) = \frac{0}{{(- 1)}^{3} - 12 {(- 1)}^{2} - 27 \cdot - 1 - 1}$

Step 14.3.2.2

Simplify the denominator.

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Step 14.3.2.2.1

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

$P' (- 1) = \frac{0}{- 1 - 12 {(- 1)}^{2} - 27 \cdot - 1 - 1}$

Step 14.3.2.2.2

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

$P' (- 1) = \frac{0}{- 1 - 12 \cdot 1 - 27 \cdot - 1 - 1}$

Step 14.3.2.2.3

Multiply $- 12$ by $1$ .

$P' (- 1) = \frac{0}{- 1 - 12 - 27 \cdot - 1 - 1}$

Step 14.3.2.2.4

Multiply $- 27$ by $- 1$ .

$P' (- 1) = \frac{0}{- 1 - 12 + 27 - 1}$

Step 14.3.2.2.5

Subtract $12$ from $- 1$ .

$P' (- 1) = \frac{0}{- 13 + 27 - 1}$

Step 14.3.2.2.6

Add $- 13$ and $27$ .

$P' (- 1) = \frac{0}{14 - 1}$

Step 14.3.2.2.7

Subtract $1$ from $14$ .

$P' (- 1) = \frac{0}{13}$

Step 14.3.2.3

Divide $0$ by $13$ .

$P' (- 1) = 0$

Step 14.3.2.4

The final answer is $0$ .

$0$

Step 14.4

Substitute any number, such as $0$ , from the interval $(- 0.03766968, 9)$ in the first derivative $\frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1}$ to check if the result is negative or positive.

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Step 14.4.1

Replace the variable $x$ with $0$ in the expression.

$P' (0) = \frac{3 ((0) + 1) ((0) - 9)}{{(0)}^{3} - 12 {(0)}^{2} - 27 \cdot 0 - 1}$

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Simplify the numerator.

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Step 14.4.2.1.1

Add $0$ and $1$ .

$P' (0) = \frac{3 \cdot (1 (0 - 9))}{0^{3} - 12 \cdot 0^{2} - 27 \cdot 0 - 1}$

Step 14.4.2.1.2

Multiply $3$ by $1$ .

$P' (0) = \frac{3 (0 - 9)}{0^{3} - 12 \cdot 0^{2} - 27 \cdot 0 - 1}$

Step 14.4.2.1.3

Subtract $9$ from $0$ .

$P' (0) = \frac{3 \cdot - 9}{0^{3} - 12 \cdot 0^{2} - 27 \cdot 0 - 1}$

Step 14.4.2.2

Simplify the denominator.

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Step 14.4.2.2.1

Raising $0$ to any positive power yields $0$ .

$P' (0) = \frac{3 \cdot - 9}{0 - 12 \cdot 0^{2} - 27 \cdot 0 - 1}$

Step 14.4.2.2.2

Raising $0$ to any positive power yields $0$ .

$P' (0) = \frac{3 \cdot - 9}{0 - 12 \cdot 0 - 27 \cdot 0 - 1}$

Step 14.4.2.2.3

Multiply $- 12$ by $0$ .

$P' (0) = \frac{3 \cdot - 9}{0 + 0 - 27 \cdot 0 - 1}$

Step 14.4.2.2.4

Multiply $- 27$ by $0$ .

$P' (0) = \frac{3 \cdot - 9}{0 + 0 + 0 - 1}$

Step 14.4.2.2.5

Add $0$ and $0$ .

$P' (0) = \frac{3 \cdot - 9}{0 + 0 - 1}$

Step 14.4.2.2.6

Add $0$ and $0$ .

$P' (0) = \frac{3 \cdot - 9}{0 - 1}$

Step 14.4.2.2.7

Subtract $1$ from $0$ .

$P' (0) = \frac{3 \cdot - 9}{- 1}$

Step 14.4.2.3

Simplify the expression.

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Step 14.4.2.3.1

Multiply $3$ by $- 9$ .

$P' (0) = \frac{- 27}{- 1}$

Step 14.4.2.3.2

Divide $- 27$ by $- 1$ .

$P' (0) = 27$

Step 14.4.2.4

The final answer is $27$ .

$27$

Step 14.5

Substitute any number, such as $11$ , from the interval $(9, \infty)$ in the first derivative $\frac{3 (x + 1) (x - 9)}{x^{3} - 12 x^{2} - 27 x - 1}$ to check if the result is negative or positive.

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Step 14.5.1

Replace the variable $x$ with $11$ in the expression.

$P' (11) = \frac{3 ((11) + 1) ((11) - 9)}{{(11)}^{3} - 12 {(11)}^{2} - 27 \cdot 11 - 1}$

Step 14.5.2

Simplify the result.

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Step 14.5.2.1

Simplify the numerator.

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Step 14.5.2.1.1

Add $11$ and $1$ .

$P' (11) = \frac{3 \cdot (12 (11 - 9))}{11^{3} - 12 \cdot 11^{2} - 27 \cdot 11 - 1}$

Step 14.5.2.1.2

Multiply $3$ by $12$ .

$P' (11) = \frac{36 (11 - 9)}{11^{3} - 12 \cdot 11^{2} - 27 \cdot 11 - 1}$

Step 14.5.2.1.3

Subtract $9$ from $11$ .

$P' (11) = \frac{36 \cdot 2}{11^{3} - 12 \cdot 11^{2} - 27 \cdot 11 - 1}$

Step 14.5.2.2

Simplify the denominator.

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Step 14.5.2.2.1

Raise $11$ to the power of $3$ .

$P' (11) = \frac{36 \cdot 2}{1331 - 12 \cdot 11^{2} - 27 \cdot 11 - 1}$

Step 14.5.2.2.2

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

$P' (11) = \frac{36 \cdot 2}{1331 - 12 \cdot 121 - 27 \cdot 11 - 1}$

Step 14.5.2.2.3

Multiply $- 12$ by $121$ .

$P' (11) = \frac{36 \cdot 2}{1331 - 1452 - 27 \cdot 11 - 1}$

Step 14.5.2.2.4

Multiply $- 27$ by $11$ .

$P' (11) = \frac{36 \cdot 2}{1331 - 1452 - 297 - 1}$

Step 14.5.2.2.5

Subtract $1452$ from $1331$ .

$P' (11) = \frac{36 \cdot 2}{- 121 - 297 - 1}$

Step 14.5.2.2.6

Subtract $297$ from $- 121$ .

$P' (11) = \frac{36 \cdot 2}{- 418 - 1}$

Step 14.5.2.2.7

Subtract $1$ from $- 418$ .

$P' (11) = \frac{36 \cdot 2}{- 419}$

Step 14.5.2.3

Simplify the expression.

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Step 14.5.2.3.1

Multiply $36$ by $2$ .

$P' (11) = \frac{72}{- 419}$

Step 14.5.2.3.2

Move the negative in front of the fraction.

$P' (11) = - \frac{72}{419}$

Step 14.5.2.4

The final answer is $- \frac{72}{419}$ .

$- \frac{72}{419}$

Step 14.6

Since the first derivative did not change signs around $x = - 1.90410133$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 14.7

Since the first derivative changed signs from negative to positive around $x = - 0.03766968$ , then $x = - 0.03766968$ is a local minimum.

$x = - 0.03766968$ is a local minimum

Step 14.8

Since the first derivative changed signs from positive to negative around $x = 9$ , then $x = 9$ is a local maximum.

$x = 9$ is a local maximum

Step 14.9

These are the local extrema for $P (x) = \ln (- x^{3} + 12 x^{2} + 27 x + 1)$ .

$x = - 0.03766968$ is a local minimum

$x = 9$ is a local maximum

$x = - 0.03766968$ is a local minimum

$x = 9$ is a local maximum

Step 15