Calculus Examples

$f (x) = \sqrt{x^{3} - 12 x^{2} + 45 x + 2}$

Step 1

Find the first derivative of the function.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3} - 12 x^{2} + 45 x + 2}$ as ${(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{3} - 12 x^{2} + 45 x + 2$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x^{3} - 12 x^{2} + 45 x + 2$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 1.2.3

Replace all occurrences of $u$ with $x^{3} - 12 x^{2} + 45 x + 2$ .

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 1.4

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{- \frac{1}{2}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 1.7.2

Combine $\frac{1}{2}$ and ${(x^{3} - 12 x^{2} + 45 x + 2)}^{- \frac{1}{2}}$ .

$\frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 1.7.3

Move ${(x^{3} - 12 x^{2} + 45 x + 2)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 1.8

By the Sum Rule, the derivative of $x^{3} - 12 x^{2} + 45 x + 2$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [45 x] + \frac{d}{d x} [2]$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [45 x] + \frac{d}{d x} [2])$

Step 1.9

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

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [45 x] + \frac{d}{d x} [2])$

Step 1.10

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

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [45 x] + \frac{d}{d x} [2])$

Step 1.11

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

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 12 (2 x) + \frac{d}{d x} [45 x] + \frac{d}{d x} [2])$

Step 1.12

Multiply $2$ by $- 12$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + \frac{d}{d x} [45 x] + \frac{d}{d x} [2])$

Step 1.13

Since $45$ is constant with respect to $x$ , the derivative of $45 x$ with respect to $x$ is $45 \frac{d}{d x} [x]$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45 \frac{d}{d x} [x] + \frac{d}{d x} [2])$

Step 1.14

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

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45 \cdot 1 + \frac{d}{d x} [2])$

Step 1.15

Multiply $45$ by $1$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45 + \frac{d}{d x} [2])$

Step 1.16

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45 + 0)$

Step 1.17

Add $3 x^{2} - 24 x + 45$ and $0$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45)$

Step 1.18

Simplify.

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Step 1.18.1

Reorder the factors of $\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45)$ .

$(3 x^{2} - 24 x + 45) \frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 1.18.2

Multiply $3 x^{2} - 24 x + 45$ by $\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ .

$\frac{3 x^{2} - 24 x + 45}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 1.18.3

Simplify the numerator.

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Step 1.18.3.1

Factor $3$ out of $3 x^{2} - 24 x + 45$ .

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Step 1.18.3.1.1

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

$\frac{3 (x^{2}) - 24 x + 45}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 1.18.3.1.2

Factor $3$ out of $- 24 x$ .

$\frac{3 (x^{2}) + 3 (- 8 x) + 45}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 1.18.3.1.3

Factor $3$ out of $45$ .

$\frac{3 x^{2} + 3 (- 8 x) + 3 \cdot 15}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 1.18.3.1.4

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

$\frac{3 (x^{2} - 8 x) + 3 \cdot 15}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 1.18.3.1.5

Factor $3$ out of $3 (x^{2} - 8 x) + 3 \cdot 15$ .

$\frac{3 (x^{2} - 8 x + 15)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 1.18.3.2

Factor $x^{2} - 8 x + 15$ using the AC method.

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Step 1.18.3.2.1

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 1.18.3.2.2

Write the factored form using these integers.

$\frac{3 ((x - 5) (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

$\frac{3 (x - 5) (x - 3)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 (x - 5) (x - 3)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [\frac{(x - 5) (x - 3)}{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}]$ .

$f'' (x) = \frac{3}{2} \cdot \frac{d}{d x} (\frac{(x - 5) (x - 3)}{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})$

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 $f (x) = (x - 5) (x - 3)$ and $g (x) = {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} \frac{d}{d x} ((x - 5) (x - 3)) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{{({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}^{2}}$

Step 2.3

Multiply the exponents in ${({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} \frac{d}{d x} ((x - 5) (x - 3)) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} \frac{d}{d x} ((x - 5) (x - 3)) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2.2

Rewrite the expression.

Step 2.4

Simplify.

Step 2.5

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 $f (x) = x - 5$ and $g (x) = x - 3$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} ((x - 5) \frac{d}{d x} (x - 3) + (x - 3) \frac{d}{d x} (x - 5)) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.6

Differentiate.

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Step 2.6.1

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

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} ((x - 5) (\frac{d}{d x} (x) + \frac{d}{d x} (- 3)) + (x - 3) \frac{d}{d x} (x - 5)) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.6.2

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

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} ((x - 5) (1 + \frac{d}{d x} (- 3)) + (x - 3) \frac{d}{d x} (x - 5)) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.6.3

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

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} ((x - 5) (1 + 0) + (x - 3) \frac{d}{d x} (x - 5)) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.6.4

Simplify the expression.

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Step 2.6.4.1

Add $1$ and $0$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} ((x - 5) \cdot 1 + (x - 3) \frac{d}{d x} (x - 5)) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.6.4.2

Multiply $x - 5$ by $1$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (x - 5 + (x - 3) \frac{d}{d x} (x - 5)) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.6.5

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (x - 5 + (x - 3) (\frac{d}{d x} (x) + \frac{d}{d x} (- 5))) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.6.6

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

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (x - 5 + (x - 3) (1 + \frac{d}{d x} (- 5))) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.6.7

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (x - 5 + (x - 3) (1 + 0)) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.6.8

Simplify by adding terms.

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Step 2.6.8.1

Add $1$ and $0$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (x - 5 + (x - 3) \cdot 1) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.6.8.2

Multiply $x - 3$ by $1$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (x - 5 + x - 3) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.6.8.3

Add $x$ and $x$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 5 - 3) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.6.8.4

Subtract $3$ from $- 5$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) \frac{d}{d x} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{3} - 12 x^{2} + 45 x + 2$ .

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Step 2.7.1

To apply the Chain Rule, set $u$ as $x^{3} - 12 x^{2} + 45 x + 2$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (x^{3} - 12 x^{2} + 45 x + 2))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.7.2

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

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (x^{3} - 12 x^{2} + 45 x + 2))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.7.3

Replace all occurrences of $u$ with $x^{3} - 12 x^{2} + 45 x + 2$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2} \cdot {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} - 1} \frac{d}{d x} (x^{3} - 12 x^{2} + 45 x + 2))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2} \cdot {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (x^{3} - 12 x^{2} + 45 x + 2))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.9

Combine $- 1$ and $\frac{2}{2}$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2} \cdot {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (x^{3} - 12 x^{2} + 45 x + 2))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.10

Combine the numerators over the common denominator.

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2} \cdot {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (x^{3} - 12 x^{2} + 45 x + 2))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.11

Simplify the numerator.

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Step 2.11.1

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2} \cdot {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1 - 2}{2}} \frac{d}{d x} (x^{3} - 12 x^{2} + 45 x + 2))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.11.2

Subtract $2$ from $1$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2} \cdot {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{- 1}{2}} \frac{d}{d x} (x^{3} - 12 x^{2} + 45 x + 2))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.12

Combine fractions.

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Step 2.12.1

Move the negative in front of the fraction.

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2} \cdot {(x^{3} - 12 x^{2} + 45 x + 2)}^{- \frac{1}{2}} \frac{d}{d x} (x^{3} - 12 x^{2} + 45 x + 2))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.12.2

Combine $\frac{1}{2}$ and ${(x^{3} - 12 x^{2} + 45 x + 2)}^{- \frac{1}{2}}$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (x^{3} - 12 x^{2} + 45 x + 2))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.12.3

Move ${(x^{3} - 12 x^{2} + 45 x + 2)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (x^{3} - 12 x^{2} + 45 x + 2))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.13

By the Sum Rule, the derivative of $x^{3} - 12 x^{2} + 45 x + 2$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [45 x] + \frac{d}{d x} [2]$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 12 x^{2}) + \frac{d}{d x} (45 x) + \frac{d}{d x} (2)))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.14

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

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} + \frac{d}{d x} (- 12 x^{2}) + \frac{d}{d x} (45 x) + \frac{d}{d x} (2)))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.15

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

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 12 \frac{d}{d x} x^{2} + \frac{d}{d x} (45 x) + \frac{d}{d x} (2)))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.16

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

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 12 (2 x) + \frac{d}{d x} (45 x) + \frac{d}{d x} (2)))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.17

Multiply $2$ by $- 12$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + \frac{d}{d x} (45 x) + \frac{d}{d x} (2)))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.18

Since $45$ is constant with respect to $x$ , the derivative of $45 x$ with respect to $x$ is $45 \frac{d}{d x} [x]$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45 \frac{d}{d x} (x) + \frac{d}{d x} (2)))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.19

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

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45 \cdot 1 + \frac{d}{d x} (2)))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.20

Multiply $45$ by $1$ .

$f'' (x) = \frac{3}{2} \cdot \frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45 + \frac{d}{d x} (2)))}{x^{3} - 12 x^{2} + 45 x + 2}$

Step 2.21

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

Step 2.22

Combine fractions.

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Step 2.22.1

Add $3 x^{2} - 24 x + 45$ and $0$ .

Step 2.22.2

Multiply $\frac{3}{2}$ by $\frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45))}{x^{3} - 12 x^{2} + 45 x + 2}$ .

$f'' (x) = \frac{3 ({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) - (x - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 (x^{3} - 12 x^{2} + 45 x + 2)}$

Step 2.23

Simplify.

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Step 2.23.1

Apply the distributive property.

$f'' (x) = \frac{3 ({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) + (- x + 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 (x^{3} - 12 x^{2} + 45 x + 2)}$

Step 2.23.2

Apply the distributive property.

$f'' (x) = \frac{3 ({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8)) + 3 ((- x + 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 (x^{3} - 12 x^{2} + 45 x + 2)}$

Step 2.23.3

Apply the distributive property.

$f'' (x) = \frac{3 ({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8)) + 3 ((- x + 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4

Simplify the numerator.

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Step 2.23.4.1

Factor $3$ out of $3 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) + 3 (- x - - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45))$ .

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Step 2.23.4.1.1

Factor $3$ out of $3 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8)$ .

$f'' (x) = \frac{3 ({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8)) + 3 (- x + 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.1.2

Factor $3$ out of $3 (- x - - 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45))$ .

$f'' (x) = \frac{3 ({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8)) + 3 (((- x + 5) (x - 3)) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.1.3

Factor $3$ out of $3 ({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8)) + 3 (((- x - - 5) (x - 3)) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45)))$ .

$f'' (x) = \frac{3 ({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x - 8) + ((- x + 5) (x - 3)) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.2

Apply the distributive property.

$f'' (x) = \frac{3 ({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x) + {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} \cdot - 8 + ((- x + 5) (x - 3)) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x + {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} \cdot - 8 + ((- x + 5) (x - 3)) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.4

Move $- 8$ to the left of ${(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 \cdot {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + ((- x + 5) (x - 3)) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.5

Multiply $- 1$ by $- 5$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x + 5) (x - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.6

Expand $(- x + 5) (x - 3)$ using the FOIL Method.

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Step 2.23.4.6.1

Apply the distributive property.

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x (x - 3) + 5 (x - 3)) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.6.2

Apply the distributive property.

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x \cdot x - x \cdot - 3 + 5 (x - 3)) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.6.3

Apply the distributive property.

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x \cdot x - x \cdot - 3 + 5 x + 5 \cdot - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.7

Simplify and combine like terms.

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Step 2.23.4.7.1

Simplify each term.

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Step 2.23.4.7.1.1

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

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Step 2.23.4.7.1.1.1

Move $x$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- (x \cdot x) - x \cdot - 3 + 5 x + 5 \cdot - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.7.1.1.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x^{2} - x \cdot - 3 + 5 x + 5 \cdot - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.7.1.2

Multiply $- 3$ by $- 1$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x^{2} + 3 x + 5 x + 5 \cdot - 3) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.7.1.3

Multiply $5$ by $- 3$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x^{2} + 3 x + 5 x - 15) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.7.2

Add $3 x$ and $5 x$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x^{2} + 8 x - 15) (\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot (3 x^{2} - 24 x + 45)))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.8

Multiply $\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ by $3 x^{2} - 24 x + 45$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x^{2} + 8 x - 15) (\frac{3 x^{2} - 24 x + 45}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.9

Simplify the numerator.

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Step 2.23.4.9.1

Factor $3$ out of $3 x^{2} - 24 x + 45$ .

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Step 2.23.4.9.1.1

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

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x^{2} + 8 x - 15) (\frac{3 (x^{2}) - 24 x + 45}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.9.1.2

Factor $3$ out of $- 24 x$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x^{2} + 8 x - 15) (\frac{3 (x^{2}) + 3 (- 8 x) + 45}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.9.1.3

Factor $3$ out of $45$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x^{2} + 8 x - 15) (\frac{3 x^{2} + 3 (- 8 x) + 3 \cdot 15}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.9.1.4

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

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x^{2} + 8 x - 15) (\frac{3 (x^{2} - 8 x) + 3 \cdot 15}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.9.1.5

Factor $3$ out of $3 (x^{2} - 8 x) + 3 \cdot 15$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x^{2} + 8 x - 15) (\frac{3 (x^{2} - 8 x + 15)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.9.2

Factor $x^{2} - 8 x + 15$ using the AC method.

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Step 2.23.4.9.2.1

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 2.23.4.9.2.2

Write the factored form using these integers.

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x^{2} + 8 x - 15) (\frac{3 ((x - 5) (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + (- x^{2} + 8 x - 15) (\frac{3 (x - 5) (x - 3)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}))}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.10

Multiply $- x^{2} + 8 x - 15$ by $\frac{3 (x - 5) (x - 3)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{(- x^{2} + 8 x - 15) (3 (x - 5) (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11

Simplify the numerator.

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Step 2.23.4.11.1

Factor by grouping.

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Step 2.23.4.11.1.1

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 $a \cdot c = - 1 \cdot - 15 = 15$ and whose sum is $b = 8$ .

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Step 2.23.4.11.1.1.1

Factor $8$ out of $8 x$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{(- x^{2} + 8 (x) - 15) \cdot (3 (x - 5) (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.1.1.2

Rewrite $8$ as $3$ plus $5$

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{(- x^{2} + (3 + 5) x - 15) \cdot (3 (x - 5) (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.1.1.3

Apply the distributive property.

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{(- x^{2} + 3 x + 5 x - 15) \cdot (3 (x - 5) (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.1.2

Factor out the greatest common factor from each group.

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Step 2.23.4.11.1.2.1

Group the first two terms and the last two terms.

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{((- x^{2} + 3 x) + 5 x - 15) \cdot (3 (x - 5) (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.1.2.2

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

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{(x (- x + 3) - 5 (- x + 3)) \cdot (3 (x - 5) (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.1.3

Factor the polynomial by factoring out the greatest common factor, $- x + 3$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{(- x + 3) (x - 5) \cdot (3 (x - 5) (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.2

Combine exponents.

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Step 2.23.4.11.2.1

Raise $x - 5$ to the power of $1$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{(- x + 3) \cdot (3 ((x - 5) (x - 5)) (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.2.2

Raise $x - 5$ to the power of $1$ .

Step 2.23.4.11.2.3

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

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{(- x + 3) \cdot (3 {(x - 5)}^{1 + 1} (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.2.4

Add $1$ and $1$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{(- x + 3) \cdot (3 {(x - 5)}^{2} (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.2.5

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

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{(- (x) + 3) \cdot (3 {(x - 5)}^{2} (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.2.6

Rewrite $3$ as $- 1 (- 3)$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{(- (x) - 1 \cdot - 3) \cdot (3 {(x - 5)}^{2} (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.2.7

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{- (x - 3) \cdot (3 {(x - 5)}^{2} (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.2.8

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{- 1 (x - 3) \cdot (3 {(x - 5)}^{2} (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.2.9

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

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{- 1 \cdot (3 {(x - 5)}^{2} ((x - 3) (x - 3)))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.2.10

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

Step 2.23.4.11.2.11

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

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{- 1 \cdot (3 {(x - 5)}^{2} {(x - 3)}^{1 + 1})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.2.12

Add $1$ and $1$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{- 1 \cdot (3 {(x - 5)}^{2} {(x - 3)}^{2})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.11.2.13

Multiply $- 1$ by $3$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} + \frac{- 3 {(x - 5)}^{2} {(x - 3)}^{2}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.12

Move the negative in front of the fraction.

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} - \frac{3 {(x - 5)}^{2} {(x - 3)}^{2}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.13

To write $2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x$ as a fraction with a common denominator, multiply by $\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{3 (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x \cdot \frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} - \frac{3 {(x - 5)}^{2} {(x - 3)}^{2}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.14

Combine $2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x$ and $\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{3 (\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} - \frac{3 {(x - 5)}^{2} {(x - 3)}^{2}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.15

Combine the numerators over the common denominator.

$f'' (x) = \frac{3 (\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}) - 3 {(x - 5)}^{2} {(x - 3)}^{2}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.16

To write $- 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{3 (\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}) - 3 {(x - 5)}^{2} {(x - 3)}^{2}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} \cdot \frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.17

Combine $- 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}$ and $\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{3 (\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}) - 3 {(x - 5)}^{2} {(x - 3)}^{2}}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} + \frac{- 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.18

Combine the numerators over the common denominator.

$f'' (x) = \frac{3 (\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}) - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19

Rewrite $\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} x (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}) - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ in a factored form.

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Step 2.23.4.19.1

Multiply ${(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}$ by ${(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.23.4.19.1.1

Move ${(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{3 (\frac{2 ({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}) x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.1.2

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

$f'' (x) = \frac{3 (\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} + \frac{1}{2}} x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.1.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{3 (\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1 + 1}{2}} x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.1.4

Add $1$ and $1$ .

$f'' (x) = \frac{3 (\frac{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{2}{2}} x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.1.5

Divide $2$ by $2$ .

$f'' (x) = \frac{3 (\frac{2 (x^{3} - 12 x^{2} + 45 x + 2) x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.2

Simplify $2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{1} x \cdot 2$ .

Step 2.23.4.19.3

Apply the distributive property.

$f'' (x) = \frac{3 (\frac{(2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2) x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.4

Simplify.

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Step 2.23.4.19.4.1

Multiply $- 12$ by $2$ .

$f'' (x) = \frac{3 (\frac{(2 x^{3} - 24 x^{2} + 2 (45 x) + 2 \cdot 2) x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.4.2

Multiply $45$ by $2$ .

$f'' (x) = \frac{3 (\frac{(2 x^{3} - 24 x^{2} + 90 x + 2 \cdot 2) x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.4.3

Multiply $2$ by $2$ .

$f'' (x) = \frac{3 (\frac{(2 x^{3} - 24 x^{2} + 90 x + 4) x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.5

Apply the distributive property.

$f'' (x) = \frac{3 (\frac{(2 x^{3} x - 24 x^{2} x + 90 x \cdot x + 4 x) \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.6

Simplify.

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Step 2.23.4.19.6.1

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

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Step 2.23.4.19.6.1.1

Move $x$ .

$f'' (x) = \frac{3 (\frac{(2 (x \cdot x^{3}) - 24 x^{2} x + 90 x \cdot x + 4 x) \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.6.1.2

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

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Step 2.23.4.19.6.1.2.1

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

Step 2.23.4.19.6.1.2.2

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

$f'' (x) = \frac{3 (\frac{(2 x^{1 + 3} - 24 x^{2} x + 90 x \cdot x + 4 x) \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.6.1.3

Add $1$ and $3$ .

$f'' (x) = \frac{3 (\frac{(2 x^{4} - 24 x^{2} x + 90 x \cdot x + 4 x) \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.6.2

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

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Step 2.23.4.19.6.2.1

Move $x$ .

$f'' (x) = \frac{3 (\frac{(2 x^{4} - 24 (x \cdot x^{2}) + 90 x \cdot x + 4 x) \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.6.2.2

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

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Step 2.23.4.19.6.2.2.1

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

Step 2.23.4.19.6.2.2.2

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

$f'' (x) = \frac{3 (\frac{(2 x^{4} - 24 x^{1 + 2} + 90 x \cdot x + 4 x) \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.6.2.3

Add $1$ and $2$ .

$f'' (x) = \frac{3 (\frac{(2 x^{4} - 24 x^{3} + 90 x \cdot x + 4 x) \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.6.3

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

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Step 2.23.4.19.6.3.1

Move $x$ .

$f'' (x) = \frac{3 (\frac{(2 x^{4} - 24 x^{3} + 90 (x \cdot x) + 4 x) \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.6.3.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{3 (\frac{(2 x^{4} - 24 x^{3} + 90 x^{2} + 4 x) \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.7

Apply the distributive property.

$f'' (x) = \frac{3 (\frac{2 x^{4} \cdot 2 - 24 x^{3} \cdot 2 + 90 x^{2} \cdot 2 + 4 x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.8

Simplify.

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Step 2.23.4.19.8.1

Multiply $2$ by $2$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 24 x^{3} \cdot 2 + 90 x^{2} \cdot 2 + 4 x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.8.2

Multiply $2$ by $- 24$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 90 x^{2} \cdot 2 + 4 x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.8.3

Multiply $2$ by $90$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 4 x \cdot 2 - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.8.4

Multiply $2$ by $4$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 {(x - 5)}^{2} {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.9

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 ((x - 5) (x - 5)) {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.10

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 2.23.4.19.10.1

Apply the distributive property.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 (x (x - 5) - 5 (x - 5)) {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.10.2

Apply the distributive property.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 (x \cdot x + x \cdot - 5 - 5 (x - 5)) {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.10.3

Apply the distributive property.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.11

Simplify and combine like terms.

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Step 2.23.4.19.11.1

Simplify each term.

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Step 2.23.4.19.11.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.11.1.2

Move $- 5$ to the left of $x$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.11.1.3

Multiply $- 5$ by $- 5$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 (x^{2} - 5 x - 5 x + 25) {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.11.2

Subtract $5 x$ from $- 5 x$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 (x^{2} - 10 x + 25) {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.12

Apply the distributive property.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x + (- 3 x^{2} - 3 (- 10 x) - 3 \cdot 25) {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.13

Simplify.

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Step 2.23.4.19.13.1

Multiply $- 10$ by $- 3$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x + (- 3 x^{2} + 30 x - 3 \cdot 25) {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.13.2

Multiply $- 3$ by $25$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x + (- 3 x^{2} + 30 x - 75) {(x - 3)}^{2} - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.14

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

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x + (- 3 x^{2} + 30 x - 75) ((x - 3) (x - 3)) - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.15

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 2.23.4.19.15.1

Apply the distributive property.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x + (- 3 x^{2} + 30 x - 75) (x (x - 3) - 3 (x - 3)) - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.15.2

Apply the distributive property.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x + (- 3 x^{2} + 30 x - 75) (x \cdot x + x \cdot - 3 - 3 (x - 3)) - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.15.3

Apply the distributive property.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x + (- 3 x^{2} + 30 x - 75) (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.16

Simplify and combine like terms.

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Step 2.23.4.19.16.1

Simplify each term.

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Step 2.23.4.19.16.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x + (- 3 x^{2} + 30 x - 75) (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.16.1.2

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

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x + (- 3 x^{2} + 30 x - 75) (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.16.1.3

Multiply $- 3$ by $- 3$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x + (- 3 x^{2} + 30 x - 75) (x^{2} - 3 x - 3 x + 9) - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.16.2

Subtract $3 x$ from $- 3 x$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x + (- 3 x^{2} + 30 x - 75) (x^{2} - 6 x + 9) - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.17

Expand $(- 3 x^{2} + 30 x - 75) (x^{2} - 6 x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{2} x^{2} - 3 x^{2} (- 6 x) - 3 x^{2} \cdot 9 + 30 x \cdot x^{2} + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18

Simplify each term.

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Step 2.23.4.19.18.1

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

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Step 2.23.4.19.18.1.1

Move $x^{2}$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 (x^{2} x^{2}) - 3 x^{2} (- 6 x) - 3 x^{2} \cdot 9 + 30 x \cdot x^{2} + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.1.2

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

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{2 + 2} - 3 x^{2} (- 6 x) - 3 x^{2} \cdot 9 + 30 x \cdot x^{2} + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.1.3

Add $2$ and $2$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} - 3 x^{2} (- 6 x) - 3 x^{2} \cdot 9 + 30 x \cdot x^{2} + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} - 3 \cdot (- 6 x^{2} x) - 3 x^{2} \cdot 9 + 30 x \cdot x^{2} + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.3

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

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Step 2.23.4.19.18.3.1

Move $x$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} - 3 \cdot (- 6 (x \cdot x^{2})) - 3 x^{2} \cdot 9 + 30 x \cdot x^{2} + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.3.2

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

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Step 2.23.4.19.18.3.2.1

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

Step 2.23.4.19.18.3.2.2

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

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} - 3 \cdot (- 6 x^{1 + 2}) - 3 x^{2} \cdot 9 + 30 x \cdot x^{2} + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.3.3

Add $1$ and $2$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} - 3 \cdot (- 6 x^{3}) - 3 x^{2} \cdot 9 + 30 x \cdot x^{2} + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.4

Multiply $- 3$ by $- 6$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 18 x^{3} - 3 x^{2} \cdot 9 + 30 x \cdot x^{2} + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.5

Multiply $9$ by $- 3$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 18 x^{3} - 27 x^{2} + 30 x \cdot x^{2} + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.6

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

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Step 2.23.4.19.18.6.1

Move $x^{2}$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 18 x^{3} - 27 x^{2} + 30 (x^{2} x) + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.6.2

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

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Step 2.23.4.19.18.6.2.1

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

Step 2.23.4.19.18.6.2.2

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

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 18 x^{3} - 27 x^{2} + 30 x^{2 + 1} + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.6.3

Add $2$ and $1$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 18 x^{3} - 27 x^{2} + 30 x^{3} + 30 x (- 6 x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.7

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 18 x^{3} - 27 x^{2} + 30 x^{3} + 30 \cdot (- 6 x \cdot x) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.8

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

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Step 2.23.4.19.18.8.1

Move $x$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 18 x^{3} - 27 x^{2} + 30 x^{3} + 30 \cdot (- 6 (x \cdot x)) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.8.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 18 x^{3} - 27 x^{2} + 30 x^{3} + 30 \cdot (- 6 x^{2}) + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.9

Multiply $30$ by $- 6$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 18 x^{3} - 27 x^{2} + 30 x^{3} - 180 x^{2} + 30 x \cdot 9 - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.10

Multiply $9$ by $30$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 18 x^{3} - 27 x^{2} + 30 x^{3} - 180 x^{2} + 270 x - 75 x^{2} - 75 (- 6 x) - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.11

Multiply $- 6$ by $- 75$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 18 x^{3} - 27 x^{2} + 30 x^{3} - 180 x^{2} + 270 x - 75 x^{2} + 450 x - 75 \cdot 9 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.18.12

Multiply $- 75$ by $9$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 18 x^{3} - 27 x^{2} + 30 x^{3} - 180 x^{2} + 270 x - 75 x^{2} + 450 x - 675 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.19

Add $18 x^{3}$ and $30 x^{3}$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 27 x^{2} - 180 x^{2} + 270 x - 75 x^{2} + 450 x - 675 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.20

Subtract $180 x^{2}$ from $- 27 x^{2}$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 207 x^{2} + 270 x - 75 x^{2} + 450 x - 675 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.21

Subtract $75 x^{2}$ from $- 207 x^{2}$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 270 x + 450 x - 675 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.22

Add $270 x$ and $450 x$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 720 x - 675 - 8 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.23

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 720 x - 675 - 8 \cdot (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.24

Multiply ${(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}$ by ${(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.23.4.19.24.1

Move ${(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 720 x - 675 - 8 \cdot (2 ({(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.24.2

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

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 720 x - 675 - 8 \cdot (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} + \frac{1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.24.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 720 x - 675 - 8 \cdot (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1 + 1}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.24.4

Add $1$ and $1$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 720 x - 675 - 8 \cdot (2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{2}{2}})}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.24.5

Divide $2$ by $2$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 720 x - 675 - 8 \cdot (2 (x^{3} - 12 x^{2} + 45 x + 2))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.25

Simplify $- 8 \cdot 2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{1}$ .

Step 2.23.4.19.26

Multiply $- 8$ by $2$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 720 x - 675 - 16 (x^{3} - 12 x^{2} + 45 x + 2)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.27

Apply the distributive property.

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 720 x - 675 - 16 x^{3} - 16 (- 12 x^{2}) - 16 (45 x) - 16 \cdot 2}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.28

Simplify.

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Step 2.23.4.19.28.1

Multiply $- 12$ by $- 16$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 720 x - 675 - 16 x^{3} + 192 x^{2} - 16 (45 x) - 16 \cdot 2}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.28.2

Multiply $45$ by $- 16$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 720 x - 675 - 16 x^{3} + 192 x^{2} - 720 x - 16 \cdot 2}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.28.3

Multiply $- 16$ by $2$ .

$f'' (x) = \frac{3 (\frac{4 x^{4} - 48 x^{3} + 180 x^{2} + 8 x - 3 x^{4} + 48 x^{3} - 282 x^{2} + 720 x - 675 - 16 x^{3} + 192 x^{2} - 720 x - 32}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.29

Subtract $3 x^{4}$ from $4 x^{4}$ .

$f'' (x) = \frac{3 (\frac{x^{4} - 48 x^{3} + 180 x^{2} + 8 x + 48 x^{3} - 282 x^{2} + 720 x - 675 - 16 x^{3} + 192 x^{2} - 720 x - 32}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.30

Add $- 48 x^{3}$ and $48 x^{3}$ .

$f'' (x) = \frac{3 (\frac{x^{4} + 180 x^{2} + 8 x + 0 - 282 x^{2} + 720 x - 675 - 16 x^{3} + 192 x^{2} - 720 x - 32}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.31

Add $x^{4}$ and $0$ .

$f'' (x) = \frac{3 (\frac{x^{4} + 180 x^{2} + 8 x - 282 x^{2} + 720 x - 675 - 16 x^{3} + 192 x^{2} - 720 x - 32}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.32

Subtract $282 x^{2}$ from $180 x^{2}$ .

$f'' (x) = \frac{3 (\frac{x^{4} - 102 x^{2} + 8 x + 720 x - 675 - 16 x^{3} + 192 x^{2} - 720 x - 32}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.33

Add $- 102 x^{2}$ and $192 x^{2}$ .

$f'' (x) = \frac{3 (\frac{x^{4} + 90 x^{2} + 8 x + 720 x - 675 - 16 x^{3} - 720 x - 32}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.34

Add $8 x$ and $720 x$ .

$f'' (x) = \frac{3 (\frac{x^{4} + 90 x^{2} + 728 x - 675 - 16 x^{3} - 720 x - 32}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.35

Subtract $720 x$ from $728 x$ .

$f'' (x) = \frac{3 (\frac{x^{4} + 90 x^{2} + 8 x - 675 - 16 x^{3} - 32}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.36

Subtract $32$ from $- 675$ .

$f'' (x) = \frac{3 (\frac{x^{4} + 90 x^{2} + 8 x - 16 x^{3} - 707}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.4.19.37

Reorder terms.

$f'' (x) = \frac{3 (\frac{x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}})}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.5

Combine terms.

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Step 2.23.5.1

Combine $3$ and $\frac{x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{\frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}}{2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2}$

Step 2.23.5.2

Multiply $- 12$ by $2$ .

$f'' (x) = \frac{\frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}}{2 x^{3} - 24 x^{2} + 2 (45 x) + 2 \cdot 2}$

Step 2.23.5.3

Multiply $45$ by $2$ .

$f'' (x) = \frac{\frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}}{2 x^{3} - 24 x^{2} + 90 x + 2 \cdot 2}$

Step 2.23.5.4

Multiply $2$ by $2$ .

$f'' (x) = \frac{\frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}}{2 x^{3} - 24 x^{2} + 90 x + 4}$

Step 2.23.5.5

Rewrite $\frac{\frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}}{2 x^{3} - 24 x^{2} + 90 x + 4}$ as a product.

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \cdot \frac{1}{2 x^{3} - 24 x^{2} + 90 x + 4}$

Step 2.23.5.6

Multiply $\frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ by $\frac{1}{2 x^{3} - 24 x^{2} + 90 x + 4}$ .

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x^{3} - 24 x^{2} + 90 x + 4)}$

Step 2.23.6

Simplify the denominator.

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Step 2.23.6.1

Factor $2$ out of $2 x^{3} - 24 x^{2} + 90 x + 4$ .

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Step 2.23.6.1.1

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

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 (x^{3}) - 24 x^{2} + 90 x + 4)}$

Step 2.23.6.1.2

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

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 (x^{3}) + 2 (- 12 x^{2}) + 90 x + 4)}$

Step 2.23.6.1.3

Factor $2$ out of $90 x$ .

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 (x^{3}) + 2 (- 12 x^{2}) + 2 (45 x) + 4)}$

Step 2.23.6.1.4

Factor $2$ out of $4$ .

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 x^{3} + 2 (- 12 x^{2}) + 2 (45 x) + 2 \cdot 2)}$

Step 2.23.6.1.5

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

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 (x^{3} - 12 x^{2}) + 2 (45 x) + 2 \cdot 2)}$

Step 2.23.6.1.6

Factor $2$ out of $2 (x^{3} - 12 x^{2}) + 2 (45 x)$ .

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 (x^{3} - 12 x^{2} + 45 x) + 2 \cdot 2)}$

Step 2.23.6.1.7

Factor $2$ out of $2 (x^{3} - 12 x^{2} + 45 x) + 2 \cdot 2$ .

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (2 (x^{3} - 12 x^{2} + 45 x + 2))}$

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} \cdot (2 (x^{3} - 12 x^{2} + 45 x + 2))}$

Step 2.23.6.2

Combine exponents.

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Step 2.23.6.2.1

Multiply $2$ by $2$ .

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{4 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}} (x^{3} - 12 x^{2} + 45 x + 2)}$

Step 2.23.6.2.2

Raise $x^{3} - 12 x^{2} + 45 x + 2$ to the power of $1$ .

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{4 ((x^{3} - 12 x^{2} + 45 x + 2) {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}})}$

Step 2.23.6.2.3

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

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{4 {(x^{3} - 12 x^{2} + 45 x + 2)}^{1 + \frac{1}{2}}}$

Step 2.23.6.2.4

Write $1$ as a fraction with a common denominator.

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{4 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 2.23.6.2.5

Combine the numerators over the common denominator.

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{4 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{2 + 1}{2}}}$

Step 2.23.6.2.6

Add $2$ and $1$ .

$f'' (x) = \frac{3 (x^{4} - 16 x^{3} + 90 x^{2} + 8 x - 707)}{4 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{3}{2}}}$

Step 3

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

$\frac{3 (x - 5) (x - 3)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} = 0$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3} - 12 x^{2} + 45 x + 2}$ as ${(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}]$

Step 4.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{3} - 12 x^{2} + 45 x + 2$ .

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Step 4.1.2.1

To apply the Chain Rule, set $u$ as $x^{3} - 12 x^{2} + 45 x + 2$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 4.1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 4.1.2.3

Replace all occurrences of $u$ with $x^{3} - 12 x^{2} + 45 x + 2$ .

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 4.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 4.1.4

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 4.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 4.1.6

Simplify the numerator.

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Step 4.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 4.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 4.1.7

Combine fractions.

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Step 4.1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(x^{3} - 12 x^{2} + 45 x + 2)}^{- \frac{1}{2}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 4.1.7.2

Combine $\frac{1}{2}$ and ${(x^{3} - 12 x^{2} + 45 x + 2)}^{- \frac{1}{2}}$ .

$\frac{{(x^{3} - 12 x^{2} + 45 x + 2)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 4.1.7.3

Move ${(x^{3} - 12 x^{2} + 45 x + 2)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} \frac{d}{d x} [x^{3} - 12 x^{2} + 45 x + 2]$

Step 4.1.8

By the Sum Rule, the derivative of $x^{3} - 12 x^{2} + 45 x + 2$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [45 x] + \frac{d}{d x} [2]$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [45 x] + \frac{d}{d x} [2])$

Step 4.1.9

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

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [45 x] + \frac{d}{d x} [2])$

Step 4.1.10

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

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [45 x] + \frac{d}{d x} [2])$

Step 4.1.11

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

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 12 (2 x) + \frac{d}{d x} [45 x] + \frac{d}{d x} [2])$

Step 4.1.12

Multiply $2$ by $- 12$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + \frac{d}{d x} [45 x] + \frac{d}{d x} [2])$

Step 4.1.13

Since $45$ is constant with respect to $x$ , the derivative of $45 x$ with respect to $x$ is $45 \frac{d}{d x} [x]$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45 \frac{d}{d x} [x] + \frac{d}{d x} [2])$

Step 4.1.14

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

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45 \cdot 1 + \frac{d}{d x} [2])$

Step 4.1.15

Multiply $45$ by $1$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45 + \frac{d}{d x} [2])$

Step 4.1.16

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45 + 0)$

Step 4.1.17

Add $3 x^{2} - 24 x + 45$ and $0$ .

$\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45)$

Step 4.1.18

Simplify.

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Step 4.1.18.1

Reorder the factors of $\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} (3 x^{2} - 24 x + 45)$ .

$(3 x^{2} - 24 x + 45) \frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 4.1.18.2

Multiply $3 x^{2} - 24 x + 45$ by $\frac{1}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ .

$\frac{3 x^{2} - 24 x + 45}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 4.1.18.3

Simplify the numerator.

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Step 4.1.18.3.1

Factor $3$ out of $3 x^{2} - 24 x + 45$ .

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Step 4.1.18.3.1.1

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

$\frac{3 (x^{2}) - 24 x + 45}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 4.1.18.3.1.2

Factor $3$ out of $- 24 x$ .

$\frac{3 (x^{2}) + 3 (- 8 x) + 45}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 4.1.18.3.1.3

Factor $3$ out of $45$ .

$\frac{3 x^{2} + 3 (- 8 x) + 3 \cdot 15}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 4.1.18.3.1.4

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

$\frac{3 (x^{2} - 8 x) + 3 \cdot 15}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 4.1.18.3.1.5

Factor $3$ out of $3 (x^{2} - 8 x) + 3 \cdot 15$ .

$\frac{3 (x^{2} - 8 x + 15)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 4.1.18.3.2

Factor $x^{2} - 8 x + 15$ using the AC method.

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Step 4.1.18.3.2.1

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 4.1.18.3.2.2

Write the factored form using these integers.

$f' (x) = \frac{3 ((x - 5) (x - 3))}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

$f' (x) = \frac{3 (x - 5) (x - 3)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{3 (x - 5) (x - 3)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$ .

$\frac{3 (x - 5) (x - 3)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{3 (x - 5) (x - 3)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{3 (x - 5) (x - 3)}{2 {(x^{3} - 12 x^{2} + 45 x + 2)}^{\frac{1}{2}}} = 0$

Step 5.2

Set the numerator equal to zero.

$3 (x - 5) (x - 3) = 0$

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 $0$ .

$x - 5 = 0$

$x - 3 = 0$

Step 5.3.2

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 5.3.2.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 5.3.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 5.3.3

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

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Step 5.3.3.1

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

$x - 3 = 0$

Step 5.3.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5.3.4

The final solution is all the values that make $3 (x - 5) (x - 3) = 0$ true.

$x = 5, 3$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{3 (x - 5) (x - 3)}{2 \sqrt{{(x^{3} - 12 x^{2} + 45 x + 2)}^{1}}}$

Step 6.1.2

Anything raised to $1$ is the base itself.

$\frac{3 (x - 5) (x - 3)}{2 \sqrt{x^{3} - 12 x^{2} + 45 x + 2}}$

Step 6.2

Set the denominator in $\frac{3 (x - 5) (x - 3)}{2 \sqrt{x^{3} - 12 x^{2} + 45 x + 2}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{x^{3} - 12 x^{2} + 45 x + 2} = 0$

Step 6.3

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

No solution

Step 6.4

Set the radicand in $\sqrt{x^{3} - 12 x^{2} + 45 x + 2}$ less than $0$ to find where the expression is undefined.

$x^{3} - 12 x^{2} + 45 x + 2 < 0$

Step 6.5

Solve for $x$ .

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Step 6.5.1

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

$x \approx - 0.04392798$

Step 6.5.2

The solution consists of all of the true intervals.

$x < - 0.04392798$

Step 6.6

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 $0$ .

$x < - 0.04392798$

$(- \infty, - 0.04392798)$

$x < - 0.04392798$

$(- \infty, - 0.04392798)$

Step 7

Critical points to evaluate.

$x = 5, 3$

Step 8

Evaluate the second derivative at $x = 5$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{3 ({(5)}^{4} - 16 {(5)}^{3} + 90 {(5)}^{2} + 8 (5) - 707)}{4 {({(5)}^{3} - 12 {(5)}^{2} + 45 (5) + 2)}^{\frac{3}{2}}}$

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 $5$ to the power of $4$ .

$\frac{3 (625 - 16 \cdot 5^{3} + 90 \cdot 5^{2} + 8 (5) - 707)}{4 {(5^{3} - 12 \cdot 5^{2} + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.1.2

Raise $5$ to the power of $3$ .

$\frac{3 (625 - 16 \cdot 125 + 90 \cdot 5^{2} + 8 (5) - 707)}{4 {(5^{3} - 12 \cdot 5^{2} + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.1.3

Multiply $- 16$ by $125$ .

$\frac{3 (625 - 2000 + 90 \cdot 5^{2} + 8 (5) - 707)}{4 {(5^{3} - 12 \cdot 5^{2} + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.1.4

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

$\frac{3 (625 - 2000 + 90 \cdot 25 + 8 (5) - 707)}{4 {(5^{3} - 12 \cdot 5^{2} + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.1.5

Multiply $90$ by $25$ .

$\frac{3 (625 - 2000 + 2250 + 8 (5) - 707)}{4 {(5^{3} - 12 \cdot 5^{2} + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.1.6

Multiply $8$ by $5$ .

$\frac{3 (625 - 2000 + 2250 + 40 - 707)}{4 {(5^{3} - 12 \cdot 5^{2} + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.1.7

Subtract $2000$ from $625$ .

$\frac{3 (- 1375 + 2250 + 40 - 707)}{4 {(5^{3} - 12 \cdot 5^{2} + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.1.8

Add $- 1375$ and $2250$ .

$\frac{3 (875 + 40 - 707)}{4 {(5^{3} - 12 \cdot 5^{2} + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.1.9

Add $875$ and $40$ .

$\frac{3 (915 - 707)}{4 {(5^{3} - 12 \cdot 5^{2} + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.1.10

Subtract $707$ from $915$ .

$\frac{3 \cdot 208}{4 {(5^{3} - 12 \cdot 5^{2} + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Raise $5$ to the power of $3$ .

$\frac{3 \cdot 208}{4 {(125 - 12 \cdot 5^{2} + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.2.1.2

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

$\frac{3 \cdot 208}{4 {(125 - 12 \cdot 25 + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.2.1.3

Multiply $- 12$ by $25$ .

$\frac{3 \cdot 208}{4 {(125 - 300 + 45 (5) + 2)}^{\frac{3}{2}}}$

Step 9.2.1.4

Multiply $45$ by $5$ .

$\frac{3 \cdot 208}{4 {(125 - 300 + 225 + 2)}^{\frac{3}{2}}}$

Step 9.2.2

Subtract $300$ from $125$ .

$\frac{3 \cdot 208}{4 {(- 175 + 225 + 2)}^{\frac{3}{2}}}$

Step 9.2.3

Add $- 175$ and $225$ .

$\frac{3 \cdot 208}{4 {(50 + 2)}^{\frac{3}{2}}}$

Step 9.2.4

Add $50$ and $2$ .

$\frac{3 \cdot 208}{4 \cdot 52^{\frac{3}{2}}}$

Step 9.3

Simplify with factoring out.

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Step 9.3.1

Multiply $3$ by $208$ .

$\frac{624}{4 \cdot 52^{\frac{3}{2}}}$

Step 9.3.2

Factor $4$ out of $624$ .

$\frac{4 \cdot 156}{4 \cdot 52^{\frac{3}{2}}}$

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $4$ out of $4 \cdot 52^{\frac{3}{2}}$ .

$\frac{4 \cdot 156}{4 (52^{\frac{3}{2}})}$

Step 9.4.2

Cancel the common factor.

$\frac{4 \cdot 156}{4 \cdot 52^{\frac{3}{2}}}$

Step 9.4.3

Rewrite the expression.

$\frac{156}{52^{\frac{3}{2}}}$

Step 10

$x = 5$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 5$ is a local minimum

Step 11

Find the y-value when $x = 5$ .

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Step 11.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = \sqrt{{(5)}^{3} - 12 {(5)}^{2} + 45 (5) + 2}$

Step 11.2

Simplify the result.

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Step 11.2.1

Raise $5$ to the power of $3$ .

$f (5) = \sqrt{125 - 12 {(5)}^{2} + 45 (5) + 2}$

Step 11.2.2

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

$f (5) = \sqrt{125 - 12 \cdot 25 + 45 (5) + 2}$

Step 11.2.3

Multiply $- 12$ by $25$ .

$f (5) = \sqrt{125 - 300 + 45 (5) + 2}$

Step 11.2.4

Multiply $45$ by $5$ .

$f (5) = \sqrt{125 - 300 + 225 + 2}$

Step 11.2.5

Subtract $300$ from $125$ .

$f (5) = \sqrt{- 175 + 225 + 2}$

Step 11.2.6

Add $- 175$ and $225$ .

$f (5) = \sqrt{50 + 2}$

Step 11.2.7

Add $50$ and $2$ .

$f (5) = \sqrt{52}$

Step 11.2.8

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 11.2.8.1

Factor $4$ out of $52$ .

$f (5) = \sqrt{4 (13)}$

Step 11.2.8.2

Rewrite $4$ as $2^{2}$ .

$f (5) = \sqrt{2^{2} \cdot 13}$

Step 11.2.9

Pull terms out from under the radical.

$f (5) = 2 \sqrt{13}$

Step 11.2.10

The final answer is $2 \sqrt{13}$ .

$y = 2 \sqrt{13}$

Step 12

Evaluate the second derivative at $x = 3$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{3 ({(3)}^{4} - 16 {(3)}^{3} + 90 {(3)}^{2} + 8 (3) - 707)}{4 {({(3)}^{3} - 12 {(3)}^{2} + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

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

$\frac{3 (81 - 16 \cdot 3^{3} + 90 \cdot 3^{2} + 8 (3) - 707)}{4 {(3^{3} - 12 \cdot 3^{2} + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.1.2

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

$\frac{3 (81 - 16 \cdot 27 + 90 \cdot 3^{2} + 8 (3) - 707)}{4 {(3^{3} - 12 \cdot 3^{2} + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.1.3

Multiply $- 16$ by $27$ .

$\frac{3 (81 - 432 + 90 \cdot 3^{2} + 8 (3) - 707)}{4 {(3^{3} - 12 \cdot 3^{2} + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.1.4

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

$\frac{3 (81 - 432 + 90 \cdot 9 + 8 (3) - 707)}{4 {(3^{3} - 12 \cdot 3^{2} + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.1.5

Multiply $90$ by $9$ .

$\frac{3 (81 - 432 + 810 + 8 (3) - 707)}{4 {(3^{3} - 12 \cdot 3^{2} + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.1.6

Multiply $8$ by $3$ .

$\frac{3 (81 - 432 + 810 + 24 - 707)}{4 {(3^{3} - 12 \cdot 3^{2} + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.1.7

Subtract $432$ from $81$ .

$\frac{3 (- 351 + 810 + 24 - 707)}{4 {(3^{3} - 12 \cdot 3^{2} + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.1.8

Add $- 351$ and $810$ .

$\frac{3 (459 + 24 - 707)}{4 {(3^{3} - 12 \cdot 3^{2} + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.1.9

Add $459$ and $24$ .

$\frac{3 (483 - 707)}{4 {(3^{3} - 12 \cdot 3^{2} + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.1.10

Subtract $707$ from $483$ .

$\frac{3 \cdot - 224}{4 {(3^{3} - 12 \cdot 3^{2} + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

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

$\frac{3 \cdot - 224}{4 {(27 - 12 \cdot 3^{2} + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.2.1.2

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

$\frac{3 \cdot - 224}{4 {(27 - 12 \cdot 9 + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.2.1.3

Multiply $- 12$ by $9$ .

$\frac{3 \cdot - 224}{4 {(27 - 108 + 45 (3) + 2)}^{\frac{3}{2}}}$

Step 13.2.1.4

Multiply $45$ by $3$ .

$\frac{3 \cdot - 224}{4 {(27 - 108 + 135 + 2)}^{\frac{3}{2}}}$

Step 13.2.2

Subtract $108$ from $27$ .

$\frac{3 \cdot - 224}{4 {(- 81 + 135 + 2)}^{\frac{3}{2}}}$

Step 13.2.3

Add $- 81$ and $135$ .

$\frac{3 \cdot - 224}{4 {(54 + 2)}^{\frac{3}{2}}}$

Step 13.2.4

Add $54$ and $2$ .

$\frac{3 \cdot - 224}{4 \cdot 56^{\frac{3}{2}}}$

Step 13.3

Simplify with factoring out.

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Step 13.3.1

Multiply $3$ by $- 224$ .

$\frac{- 672}{4 \cdot 56^{\frac{3}{2}}}$

Step 13.3.2

Factor $4$ out of $- 672$ .

$\frac{4 \cdot - 168}{4 \cdot 56^{\frac{3}{2}}}$

Step 13.4

Cancel the common factors.

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Step 13.4.1

Factor $4$ out of $4 \cdot 56^{\frac{3}{2}}$ .

$\frac{4 \cdot - 168}{4 (56^{\frac{3}{2}})}$

Step 13.4.2

Cancel the common factor.

$\frac{4 \cdot - 168}{4 \cdot 56^{\frac{3}{2}}}$

Step 13.4.3

Rewrite the expression.

$\frac{- 168}{56^{\frac{3}{2}}}$

Step 13.5

Move the negative in front of the fraction.

$- \frac{168}{56^{\frac{3}{2}}}$

Step 14

$x = 3$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 3$ is a local maximum

Step 15

Find the y-value when $x = 3$ .

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Step 15.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \sqrt{{(3)}^{3} - 12 {(3)}^{2} + 45 (3) + 2}$

Step 15.2

Simplify the result.

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Step 15.2.1

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

$f (3) = \sqrt{27 - 12 {(3)}^{2} + 45 (3) + 2}$

Step 15.2.2

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

$f (3) = \sqrt{27 - 12 \cdot 9 + 45 (3) + 2}$

Step 15.2.3

Multiply $- 12$ by $9$ .

$f (3) = \sqrt{27 - 108 + 45 (3) + 2}$

Step 15.2.4

Multiply $45$ by $3$ .

$f (3) = \sqrt{27 - 108 + 135 + 2}$

Step 15.2.5

Subtract $108$ from $27$ .

$f (3) = \sqrt{- 81 + 135 + 2}$

Step 15.2.6

Add $- 81$ and $135$ .

$f (3) = \sqrt{54 + 2}$

Step 15.2.7

Add $54$ and $2$ .

$f (3) = \sqrt{56}$

Step 15.2.8

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 15.2.8.1

Factor $4$ out of $56$ .

$f (3) = \sqrt{4 (14)}$

Step 15.2.8.2

Rewrite $4$ as $2^{2}$ .

$f (3) = \sqrt{2^{2} \cdot 14}$

Step 15.2.9

Pull terms out from under the radical.

$f (3) = 2 \sqrt{14}$

Step 15.2.10

The final answer is $2 \sqrt{14}$ .

$y = 2 \sqrt{14}$

Step 16

These are the local extrema for $f (x) = \sqrt{x^{3} - 12 x^{2} + 45 x + 2}$ .

$(5, 2 \sqrt{13})$ is a local minima

$(3, 2 \sqrt{14})$ is a local maxima

Step 17