Calculus Examples

$y = \frac{x^{2} + 1}{x^{2} - 9}$

Step 1

Write $y = \frac{x^{2} + 1}{x^{2} - 9}$ as a function.

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

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} + 1$ and $g (x) = x^{2} - 9$ .

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

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

$f' (x) = \frac{(x^{2} - 9) (2 x + 0) - (x^{2} + 1) \frac{d}{d x} (x^{2} - 9)}{{(x^{2} - 9)}^{2}}$

Step 2.1.2.4

Simplify the expression.

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Step 2.1.2.4.1

Add $2 x$ and $0$ .

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

Step 2.1.2.4.2

Move $2$ to the left of $x^{2} - 9$ .

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

Step 2.1.2.5

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

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

Step 2.1.2.6

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

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

Step 2.1.2.7

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

$f' (x) = \frac{2 (x^{2} - 9) x - (x^{2} + 1) (2 x + 0)}{{(x^{2} - 9)}^{2}}$

Step 2.1.2.8

Simplify the expression.

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Step 2.1.2.8.1

Add $2 x$ and $0$ .

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

Step 2.1.2.8.2

Multiply $2$ by $- 1$ .

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

Step 2.1.3

Simplify.

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Step 2.1.3.1

Apply the distributive property.

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

Step 2.1.3.2

Apply the distributive property.

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

Step 2.1.3.3

Apply the distributive property.

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

Step 2.1.3.4

Apply the distributive property.

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

Step 2.1.3.5

Simplify the numerator.

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Step 2.1.3.5.1

Combine the opposite terms in $2 x^{2} x + 2 \cdot - 9 x - 2 x^{2} x - 2 \cdot 1 x$ .

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Step 2.1.3.5.1.1

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

$f' (x) = \frac{2 \cdot (- 9 x) + 0 - 2 \cdot (1 x)}{{(x^{2} - 9)}^{2}}$

Step 2.1.3.5.1.2

Add $2 \cdot - 9 x$ and $0$ .

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

Step 2.1.3.5.2

Simplify each term.

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Step 2.1.3.5.2.1

Multiply $2$ by $- 9$ .

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

Step 2.1.3.5.2.2

Multiply $- 2$ by $1$ .

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

Step 2.1.3.5.3

Subtract $2 x$ from $- 18 x$ .

$f' (x) = \frac{- 20 x}{{(x^{2} - 9)}^{2}}$

Step 2.1.3.6

Move the negative in front of the fraction.

$f' (x) = - \frac{20 x}{{(x^{2} - 9)}^{2}}$

Step 2.1.3.7

Simplify the denominator.

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Step 2.1.3.7.1

Rewrite $9$ as $3^{2}$ .

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

Step 2.1.3.7.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Step 2.1.3.7.3

Apply the product rule to $(x + 3) (x - 3)$ .

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

Step 2.2

Find the second derivative.

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Step 2.2.1

Since $- 20$ is constant with respect to $x$ , the derivative of $- \frac{20 x}{{(x + 3)}^{2} {(x - 3)}^{2}}$ with respect to $x$ is $- 20 \frac{d}{d x} [\frac{x}{{(x + 3)}^{2} {(x - 3)}^{2}}]$ .

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

Step 2.2.2

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

Step 2.2.3

Differentiate using the Power Rule.

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Step 2.2.3.1

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

$f'' (x) = - 20 \frac{{(x + 3)}^{2} {(x - 3)}^{2} \cdot 1 - x \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.3.2

Multiply ${(x + 3)}^{2}$ by $1$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

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Step 2.2.5.1

To apply the Chain Rule, set $u_{1}$ as $x - 3$ .

$f'' (x) = - 20 \frac{{(x + 3)}^{2} {(x - 3)}^{2} - x ({(x + 3)}^{2} (\frac{d}{d u (1)} ({u_{1}}^{2}) \frac{d}{d x} (x - 3)) + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.5.2

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

$f'' (x) = - 20 \frac{{(x + 3)}^{2} {(x - 3)}^{2} - x ({(x + 3)}^{2} (2 u_{1} \frac{d}{d x} (x - 3)) + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.5.3

Replace all occurrences of $u_{1}$ with $x - 3$ .

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

Step 2.2.6

Differentiate.

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Step 2.2.6.1

Move $2$ to the left of ${(x + 3)}^{2}$ .

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

Step 2.2.6.2

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

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

Step 2.2.6.3

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

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

Step 2.2.6.4

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

$f'' (x) = - 20 \frac{{(x + 3)}^{2} {(x - 3)}^{2} - x (2 {(x + 3)}^{2} (x - 3) (1 + 0) + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.6.5

Simplify the expression.

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Step 2.2.6.5.1

Add $1$ and $0$ .

$f'' (x) = - 20 \frac{{(x + 3)}^{2} {(x - 3)}^{2} - x (2 {(x + 3)}^{2} (x - 3) \cdot 1 + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.6.5.2

Multiply $2$ by $1$ .

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

Step 2.2.7

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

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Step 2.2.7.1

To apply the Chain Rule, set $u_{2}$ as $x + 3$ .

$f'' (x) = - 20 \frac{{(x + 3)}^{2} {(x - 3)}^{2} - x (2 {(x + 3)}^{2} (x - 3) + {(x - 3)}^{2} (\frac{d}{d u (2)} ({u_{2}}^{2}) \frac{d}{d x} (x + 3)))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.7.2

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

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

Step 2.2.7.3

Replace all occurrences of $u_{2}$ with $x + 3$ .

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

Step 2.2.8

Differentiate.

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Step 2.2.8.1

Move $2$ to the left of ${(x - 3)}^{2}$ .

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

Step 2.2.8.2

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

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

Step 2.2.8.3

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

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

Step 2.2.8.4

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

$f'' (x) = - 20 \frac{{(x + 3)}^{2} {(x - 3)}^{2} - x (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3) (1 + 0))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.8.5

Combine fractions.

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Step 2.2.8.5.1

Add $1$ and $0$ .

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

Step 2.2.8.5.2

Multiply $2$ by $1$ .

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

Step 2.2.8.5.3

Combine $- 20$ and $\frac{{(x + 3)}^{2} {(x - 3)}^{2} - x (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$ .

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

Step 2.2.8.5.4

Move the negative in front of the fraction.

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

Step 2.2.9

Simplify.

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Step 2.2.9.1

Apply the product rule to ${(x + 3)}^{2} {(x - 3)}^{2}$ .

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

Step 2.2.9.2

Apply the distributive property.

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

Step 2.2.9.3

Apply the distributive property.

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

Step 2.2.9.4

Simplify the numerator.

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Step 2.2.9.4.1

Factor $20 (x + 3) (x - 3)$ out of $20 {(x + 3)}^{2} {(x - 3)}^{2} + 20 (- x (2 {(x + 3)}^{2} (x - 3))) + 20 (- x (2 {(x - 3)}^{2} (x + 3)))$ .

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Step 2.2.9.4.1.1

Factor $20 (x + 3) (x - 3)$ out of $20 {(x + 3)}^{2} {(x - 3)}^{2}$ .

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

Step 2.2.9.4.1.2

Factor $20 (x + 3) (x - 3)$ out of $20 (- x (2 {(x + 3)}^{2} (x - 3)))$ .

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

Step 2.2.9.4.1.3

Factor $20 (x + 3) (x - 3)$ out of $20 (- x (2 {(x - 3)}^{2} (x + 3)))$ .

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

Step 2.2.9.4.1.4

Factor $20 (x + 3) (x - 3)$ out of $20 (x + 3) (x - 3) ((x + 3) (x - 3)) + 20 (x + 3) (x - 3) (- x (2 (x + 3)))$ .

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

Step 2.2.9.4.1.5

Factor $20 (x + 3) (x - 3)$ out of $20 (x + 3) (x - 3) ((x + 3) (x - 3) - x (2 (x + 3))) + 20 (x + 3) (x - 3) (- x (2 (x - 3)))$ .

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

Step 2.2.9.4.2

Combine exponents.

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Step 2.2.9.4.2.1

Multiply $2$ by $- 1$ .

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

Step 2.2.9.4.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.9.4.3

Simplify each term.

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Step 2.2.9.4.3.1

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

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Step 2.2.9.4.3.1.1

Apply the distributive property.

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

Step 2.2.9.4.3.1.2

Apply the distributive property.

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

Step 2.2.9.4.3.1.3

Apply the distributive property.

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

Step 2.2.9.4.3.2

Combine the opposite terms in $x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3$ .

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Step 2.2.9.4.3.2.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

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

Step 2.2.9.4.3.2.2

Add $- 3 x$ and $3 x$ .

$f'' (x) = - \frac{20 (x + 3) (x - 3) (x \cdot x + 0 + 3 \cdot - 3 - 2 x (x + 3) - 2 x (x - 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.3.2.3

Add $x \cdot x$ and $0$ .

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

Step 2.2.9.4.3.3

Simplify each term.

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Step 2.2.9.4.3.3.1

Multiply $x$ by $x$ .

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

Step 2.2.9.4.3.3.2

Multiply $3$ by $- 3$ .

$f'' (x) = - \frac{20 (x + 3) (x - 3) (x^{2} - 9 - 2 x (x + 3) - 2 x (x - 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.3.4

Apply the distributive property.

$f'' (x) = - \frac{20 (x + 3) (x - 3) (x^{2} - 9 - 2 x \cdot x - 2 x \cdot 3 - 2 x (x - 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.3.5

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

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Step 2.2.9.4.3.5.1

Move $x$ .

$f'' (x) = - \frac{20 (x + 3) (x - 3) (x^{2} - 9 - 2 (x \cdot x) - 2 x \cdot 3 - 2 x (x - 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.3.5.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{20 (x + 3) (x - 3) (x^{2} - 9 - 2 x^{2} - 2 x \cdot 3 - 2 x (x - 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.3.6

Multiply $3$ by $- 2$ .

$f'' (x) = - \frac{20 (x + 3) (x - 3) (x^{2} - 9 - 2 x^{2} - 6 x - 2 x (x - 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.3.7

Apply the distributive property.

$f'' (x) = - \frac{20 (x + 3) (x - 3) (x^{2} - 9 - 2 x^{2} - 6 x - 2 x \cdot x - 2 x \cdot - 3)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.3.8

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

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Step 2.2.9.4.3.8.1

Move $x$ .

$f'' (x) = - \frac{20 (x + 3) (x - 3) (x^{2} - 9 - 2 x^{2} - 6 x - 2 (x \cdot x) - 2 x \cdot - 3)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.3.8.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{20 (x + 3) (x - 3) (x^{2} - 9 - 2 x^{2} - 6 x - 2 x^{2} - 2 x \cdot - 3)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.3.9

Multiply $- 3$ by $- 2$ .

$f'' (x) = - \frac{20 (x + 3) (x - 3) (x^{2} - 9 - 2 x^{2} - 6 x - 2 x^{2} + 6 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.4

Combine the opposite terms in $x^{2} - 9 - 2 x^{2} - 6 x - 2 x^{2} + 6 x$ .

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Step 2.2.9.4.4.1

Add $- 6 x$ and $6 x$ .

$f'' (x) = - \frac{20 (x + 3) (x - 3) (x^{2} - 9 - 2 x^{2} - 2 x^{2} + 0)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.4.2

Add $x^{2} - 9 - 2 x^{2} - 2 x^{2}$ and $0$ .

$f'' (x) = - \frac{20 (x + 3) (x - 3) (x^{2} - 9 - 2 x^{2} - 2 x^{2})}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.5

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

$f'' (x) = - \frac{20 (x + 3) (x - 3) (- x^{2} - 9 - 2 x^{2})}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.6

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

$f'' (x) = - \frac{20 (x + 3) (x - 3) (- 3 x^{2} - 9)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.7

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

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Step 2.2.9.4.7.1

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

$f'' (x) = - \frac{20 (x + 3) (x - 3) (3 (- x^{2}) - 9)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.4.7.2

Factor $3$ out of $- 9$ .

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

Step 2.2.9.4.7.3

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

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

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

Step 2.2.9.4.8

Multiply $3$ by $20$ .

$f'' (x) = - \frac{60 (x + 3) (x - 3) (- x^{2} - 3)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.5

Combine terms.

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Step 2.2.9.5.1

Multiply the exponents in ${({(x + 3)}^{2})}^{2}$ .

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Step 2.2.9.5.1.1

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

$f'' (x) = - \frac{60 (x + 3) (x - 3) (- x^{2} - 3)}{{(x + 3)}^{2 \cdot 2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.5.1.2

Multiply $2$ by $2$ .

$f'' (x) = - \frac{60 (x + 3) (x - 3) (- x^{2} - 3)}{{(x + 3)}^{4} {({(x - 3)}^{2})}^{2}}$

Step 2.2.9.5.2

Multiply the exponents in ${({(x - 3)}^{2})}^{2}$ .

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Step 2.2.9.5.2.1

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

$f'' (x) = - \frac{60 (x + 3) (x - 3) (- x^{2} - 3)}{{(x + 3)}^{4} {(x - 3)}^{2 \cdot 2}}$

Step 2.2.9.5.2.2

Multiply $2$ by $2$ .

$f'' (x) = - \frac{60 (x + 3) (x - 3) (- x^{2} - 3)}{{(x + 3)}^{4} {(x - 3)}^{4}}$

Step 2.2.9.5.3

Cancel the common factor of $x + 3$ and ${(x + 3)}^{4}$ .

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Step 2.2.9.5.3.1

Factor $x + 3$ out of $60 (x + 3) (x - 3) (- x^{2} - 3)$ .

$f'' (x) = - \frac{(x + 3) ((60 (x - 3)) (- x^{2} - 3))}{{(x + 3)}^{4} {(x - 3)}^{4}}$

Step 2.2.9.5.3.2

Cancel the common factors.

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Step 2.2.9.5.3.2.1

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

$f'' (x) = - \frac{(x + 3) ((60 (x - 3)) (- x^{2} - 3))}{(x + 3) ({(x + 3)}^{3} {(x - 3)}^{4})}$

Step 2.2.9.5.3.2.2

Cancel the common factor.

$f'' (x) = - \frac{(x + 3) ((60 (x - 3)) (- x^{2} - 3))}{(x + 3) ({(x + 3)}^{3} {(x - 3)}^{4})}$

Step 2.2.9.5.3.2.3

Rewrite the expression.

$f'' (x) = - \frac{(60 (x - 3)) (- x^{2} - 3)}{{(x + 3)}^{3} {(x - 3)}^{4}}$

Step 2.2.9.5.4

Cancel the common factor of $x - 3$ and ${(x - 3)}^{4}$ .

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Step 2.2.9.5.4.1

Factor $x - 3$ out of $60 (x - 3) (- x^{2} - 3)$ .

$f'' (x) = - \frac{(x - 3) (60 (- x^{2} - 3))}{{(x + 3)}^{3} {(x - 3)}^{4}}$

Step 2.2.9.5.4.2

Cancel the common factors.

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Step 2.2.9.5.4.2.1

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

$f'' (x) = - \frac{(x - 3) (60 (- x^{2} - 3))}{(x - 3) ({(x + 3)}^{3} {(x - 3)}^{3})}$

Step 2.2.9.5.4.2.2

Cancel the common factor.

$f'' (x) = - \frac{(x - 3) (60 (- x^{2} - 3))}{(x - 3) ({(x + 3)}^{3} {(x - 3)}^{3})}$

Step 2.2.9.5.4.2.3

Rewrite the expression.

$f'' (x) = - \frac{60 (- x^{2} - 3)}{{(x + 3)}^{3} {(x - 3)}^{3}}$

Step 2.2.9.6

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

$f'' (x) = - \frac{60 (- (x^{2}) - 3)}{{(x + 3)}^{3} {(x - 3)}^{3}}$

Step 2.2.9.7

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

$f'' (x) = - \frac{60 (- (x^{2}) - 1 \cdot 3)}{{(x + 3)}^{3} {(x - 3)}^{3}}$

Step 2.2.9.8

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

$f'' (x) = - \frac{60 (- (x^{2} + 3))}{{(x + 3)}^{3} {(x - 3)}^{3}}$

Step 2.2.9.9

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

$f'' (x) = - \frac{60 (- 1 (x^{2} + 3))}{{(x + 3)}^{3} {(x - 3)}^{3}}$

Step 2.2.9.10

Move the negative in front of the fraction.

$f'' (x) = \frac{60 (x^{2} + 3)}{{(x + 3)}^{3} {(x - 3)}^{3}}$

Step 2.2.9.11

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{60 (x^{2} + 3)}{{(x + 3)}^{3} {(x - 3)}^{3}})$

Step 2.2.9.12

Multiply $\frac{60 (x^{2} + 3)}{{(x + 3)}^{3} {(x - 3)}^{3}}$ by $1$ .

$f'' (x) = \frac{60 (x^{2} + 3)}{{(x + 3)}^{3} {(x - 3)}^{3}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{60 (x^{2} + 3)}{{(x + 3)}^{3} {(x - 3)}^{3}}$ .

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

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{60 (x^{2} + 3)}{{(x + 3)}^{3} {(x - 3)}^{3}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$\frac{60 (x^{2} + 3)}{{(x + 3)}^{3} {(x - 3)}^{3}} = 0$

Step 3.2

Set the numerator equal to zero.

$60 (x^{2} + 3) = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

Divide each term in $60 (x^{2} + 3) = 0$ by $60$ and simplify.

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Step 3.3.1.1

Divide each term in $60 (x^{2} + 3) = 0$ by $60$ .

$\frac{60 (x^{2} + 3)}{60} = \frac{0}{60}$

Step 3.3.1.2

Simplify the left side.

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Step 3.3.1.2.1

Cancel the common factor of $60$ .

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Step 3.3.1.2.1.1

Cancel the common factor.

$\frac{60 (x^{2} + 3)}{60} = \frac{0}{60}$

Step 3.3.1.2.1.2

Divide $x^{2} + 3$ by $1$ .

$x^{2} + 3 = \frac{0}{60}$

Step 3.3.1.3

Simplify the right side.

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Step 3.3.1.3.1

Divide $0$ by $60$ .

$x^{2} + 3 = 0$

Step 3.3.2

Subtract $3$ from both sides of the equation.

$x^{2} = - 3$

Step 3.3.3

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

$x = \pm \sqrt{- 3}$

Step 3.3.4

Simplify $\pm \sqrt{- 3}$ .

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Step 3.3.4.1

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

$x = \pm \sqrt{- 1 (3)}$

Step 3.3.4.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{3}$

Step 3.3.4.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{3}$

Step 3.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{3}$

Step 3.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{3}$

Step 3.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{3}, - i \sqrt{3}$

Step 4

No values found that can make the second derivative equal to $0$ .

No Inflection Points