Calculus Examples

$\frac{x}{1 - x^{2}}$

Step 1

Write $\frac{x}{1 - x^{2}}$ as a function.

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

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

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 $f (x) = x$ and $g (x) = 1 - x^{2}$ .

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

Step 2.1.1.2

Differentiate.

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Step 2.1.1.2.1

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

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

Step 2.1.1.2.2

Multiply $1 - x^{2}$ by $1$ .

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

Step 2.1.1.2.3

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

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

Step 2.1.1.2.4

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

$\frac{1 - x^{2} - x (0 + \frac{d}{d x} [- x^{2}])}{{(1 - x^{2})}^{2}}$

Step 2.1.1.2.5

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 2.1.1.2.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 2.1.1.2.7

Multiply.

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Step 2.1.1.2.7.1

Multiply $- 1$ by $- 1$ .

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

Step 2.1.1.2.7.2

Multiply $x$ by $1$ .

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

Step 2.1.1.2.8

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

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

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

Step 2.1.1.5

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

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

Step 2.1.1.6

Add $1$ and $1$ .

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

Step 2.1.1.7

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

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

Step 2.1.1.8

Reorder terms.

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

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

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

Step 2.1.2.2

Differentiate.

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Step 2.1.2.2.1

Multiply the exponents in ${({(- x^{2} + 1)}^{2})}^{2}$ .

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Step 2.1.2.2.1.1

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

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

Step 2.1.2.2.1.2

Multiply $2$ by $2$ .

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

Step 2.1.2.2.2

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

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

Step 2.1.2.2.3

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

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

Step 2.1.2.2.4

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

$\frac{{(- x^{2} + 1)}^{2} (2 x + 0) - (x^{2} + 1) \frac{d}{d x} [{(- x^{2} + 1)}^{2}]}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.2.5

Simplify the expression.

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Step 2.1.2.2.5.1

Add $2 x$ and $0$ .

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

Step 2.1.2.2.5.2

Move $2$ to the left of ${(- x^{2} + 1)}^{2}$ .

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

Step 2.1.2.3

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 $g (x) = - x^{2} + 1$ .

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Step 2.1.2.3.1

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

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

Step 2.1.2.3.2

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

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

Step 2.1.2.3.3

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

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

Step 2.1.2.4

Differentiate.

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Step 2.1.2.4.1

Multiply $2$ by $- 1$ .

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

Step 2.1.2.4.2

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

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

Step 2.1.2.4.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 2.1.2.4.4

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

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

Step 2.1.2.4.5

Multiply $2$ by $- 1$ .

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

Step 2.1.2.4.6

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

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

Step 2.1.2.4.7

Simplify the expression.

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Step 2.1.2.4.7.1

Add $- 2 x$ and $0$ .

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

Step 2.1.2.4.7.2

Multiply $- 2$ by $- 2$ .

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

Step 2.1.2.5

Simplify.

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Step 2.1.2.5.1

Apply the distributive property.

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

Step 2.1.2.5.2

Apply the distributive property.

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

Step 2.1.2.5.3

Simplify the numerator.

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Step 2.1.2.5.3.1

Simplify each term.

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Step 2.1.2.5.3.1.1

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

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

Step 2.1.2.5.3.1.2

Expand $(- x^{2} + 1) (- x^{2} + 1)$ using the FOIL Method.

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Step 2.1.2.5.3.1.2.1

Apply the distributive property.

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

Step 2.1.2.5.3.1.2.2

Apply the distributive property.

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

Step 2.1.2.5.3.1.2.3

Apply the distributive property.

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

Step 2.1.2.5.3.1.3

Simplify and combine like terms.

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Step 2.1.2.5.3.1.3.1

Simplify each term.

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Step 2.1.2.5.3.1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.1.2.5.3.1.3.1.2

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

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Step 2.1.2.5.3.1.3.1.2.1

Move $x^{2}$ .

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

Step 2.1.2.5.3.1.3.1.2.2

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

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

Step 2.1.2.5.3.1.3.1.2.3

Add $2$ and $2$ .

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

Step 2.1.2.5.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

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

Step 2.1.2.5.3.1.3.1.4

Multiply $x^{4}$ by $1$ .

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

Step 2.1.2.5.3.1.3.1.5

Multiply $- 1$ by $1$ .

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

Step 2.1.2.5.3.1.3.1.6

Multiply $- x^{2}$ by $1$ .

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

Step 2.1.2.5.3.1.3.1.7

Multiply $1$ by $1$ .

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

Step 2.1.2.5.3.1.3.2

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

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

Step 2.1.2.5.3.1.4

Apply the distributive property.

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

Step 2.1.2.5.3.1.5

Simplify.

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Step 2.1.2.5.3.1.5.1

Multiply $- 2$ by $2$ .

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

Step 2.1.2.5.3.1.5.2

Multiply $2$ by $1$ .

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

Step 2.1.2.5.3.1.6

Apply the distributive property.

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

Step 2.1.2.5.3.1.7

Simplify.

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Step 2.1.2.5.3.1.7.1

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 2.1.2.5.3.1.7.1.1

Move $x$ .

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

Step 2.1.2.5.3.1.7.1.2

Multiply $x$ by $x^{4}$ .

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Step 2.1.2.5.3.1.7.1.2.1

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

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

Step 2.1.2.5.3.1.7.1.2.2

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

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

Step 2.1.2.5.3.1.7.1.3

Add $1$ and $4$ .

$\frac{2 x^{5} - 4 x^{2} x + 2 x + (4 x^{2} + 4 \cdot 1) (- x^{2} x + 1 x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.7.2

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

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Step 2.1.2.5.3.1.7.2.1

Move $x$ .

$\frac{2 x^{5} - 4 (x \cdot x^{2}) + 2 x + (4 x^{2} + 4 \cdot 1) (- x^{2} x + 1 x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.7.2.2

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

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Step 2.1.2.5.3.1.7.2.2.1

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

$\frac{2 x^{5} - 4 (x^{1} x^{2}) + 2 x + (4 x^{2} + 4 \cdot 1) (- x^{2} x + 1 x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.7.2.2.2

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

$\frac{2 x^{5} - 4 x^{1 + 2} + 2 x + (4 x^{2} + 4 \cdot 1) (- x^{2} x + 1 x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.7.2.3

Add $1$ and $2$ .

$\frac{2 x^{5} - 4 x^{3} + 2 x + (4 x^{2} + 4 \cdot 1) (- x^{2} x + 1 x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.8

Multiply $4$ by $1$ .

$\frac{2 x^{5} - 4 x^{3} + 2 x + (4 x^{2} + 4) (- x^{2} x + 1 x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.9

Simplify each term.

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Step 2.1.2.5.3.1.9.1

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

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Step 2.1.2.5.3.1.9.1.1

Move $x$ .

$\frac{2 x^{5} - 4 x^{3} + 2 x + (4 x^{2} + 4) (- (x \cdot x^{2}) + 1 x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.9.1.2

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

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Step 2.1.2.5.3.1.9.1.2.1

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

$\frac{2 x^{5} - 4 x^{3} + 2 x + (4 x^{2} + 4) (- (x^{1} x^{2}) + 1 x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.9.1.2.2

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

$\frac{2 x^{5} - 4 x^{3} + 2 x + (4 x^{2} + 4) (- x^{1 + 2} + 1 x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.9.1.3

Add $1$ and $2$ .

$\frac{2 x^{5} - 4 x^{3} + 2 x + (4 x^{2} + 4) (- x^{3} + 1 x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.9.2

Multiply $x$ by $1$ .

$\frac{2 x^{5} - 4 x^{3} + 2 x + (4 x^{2} + 4) (- x^{3} + x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.10

Expand $(4 x^{2} + 4) (- x^{3} + x)$ using the FOIL Method.

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Step 2.1.2.5.3.1.10.1

Apply the distributive property.

$\frac{2 x^{5} - 4 x^{3} + 2 x + 4 x^{2} (- x^{3} + x) + 4 (- x^{3} + x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.10.2

Apply the distributive property.

$\frac{2 x^{5} - 4 x^{3} + 2 x + 4 x^{2} (- x^{3}) + 4 x^{2} x + 4 (- x^{3} + x)}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.10.3

Apply the distributive property.

$\frac{2 x^{5} - 4 x^{3} + 2 x + 4 x^{2} (- x^{3}) + 4 x^{2} x + 4 (- x^{3}) + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.11

Simplify and combine like terms.

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Step 2.1.2.5.3.1.11.1

Simplify each term.

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Step 2.1.2.5.3.1.11.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{5} - 4 x^{3} + 2 x + 4 \cdot - 1 x^{2} x^{3} + 4 x^{2} x + 4 (- x^{3}) + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.11.1.2

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 2.1.2.5.3.1.11.1.2.1

Move $x^{3}$ .

$\frac{2 x^{5} - 4 x^{3} + 2 x + 4 \cdot - 1 (x^{3} x^{2}) + 4 x^{2} x + 4 (- x^{3}) + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.11.1.2.2

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

$\frac{2 x^{5} - 4 x^{3} + 2 x + 4 \cdot - 1 x^{3 + 2} + 4 x^{2} x + 4 (- x^{3}) + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.11.1.2.3

Add $3$ and $2$ .

$\frac{2 x^{5} - 4 x^{3} + 2 x + 4 \cdot - 1 x^{5} + 4 x^{2} x + 4 (- x^{3}) + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.11.1.3

Multiply $4$ by $- 1$ .

$\frac{2 x^{5} - 4 x^{3} + 2 x - 4 x^{5} + 4 x^{2} x + 4 (- x^{3}) + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.11.1.4

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

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Step 2.1.2.5.3.1.11.1.4.1

Move $x$ .

$\frac{2 x^{5} - 4 x^{3} + 2 x - 4 x^{5} + 4 (x \cdot x^{2}) + 4 (- x^{3}) + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.11.1.4.2

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

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Step 2.1.2.5.3.1.11.1.4.2.1

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

$\frac{2 x^{5} - 4 x^{3} + 2 x - 4 x^{5} + 4 (x^{1} x^{2}) + 4 (- x^{3}) + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.11.1.4.2.2

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

$\frac{2 x^{5} - 4 x^{3} + 2 x - 4 x^{5} + 4 x^{1 + 2} + 4 (- x^{3}) + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.11.1.4.3

Add $1$ and $2$ .

$\frac{2 x^{5} - 4 x^{3} + 2 x - 4 x^{5} + 4 x^{3} + 4 (- x^{3}) + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.11.1.5

Multiply $- 1$ by $4$ .

$\frac{2 x^{5} - 4 x^{3} + 2 x - 4 x^{5} + 4 x^{3} - 4 x^{3} + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.11.2

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

$\frac{2 x^{5} - 4 x^{3} + 2 x - 4 x^{5} + 0 + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.1.11.3

Add $- 4 x^{5}$ and $0$ .

$\frac{2 x^{5} - 4 x^{3} + 2 x - 4 x^{5} + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.2

Subtract $4 x^{5}$ from $2 x^{5}$ .

$\frac{- 2 x^{5} - 4 x^{3} + 2 x + 4 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.3.3

Add $2 x$ and $4 x$ .

$\frac{- 2 x^{5} - 4 x^{3} + 6 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.4

Simplify the numerator.

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Step 2.1.2.5.4.1

Factor $2 x$ out of $- 2 x^{5} - 4 x^{3} + 6 x$ .

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Step 2.1.2.5.4.1.1

Factor $2 x$ out of $- 2 x^{5}$ .

$\frac{2 x (- x^{4}) - 4 x^{3} + 6 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.4.1.2

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

$\frac{2 x (- x^{4}) + 2 x (- 2 x^{2}) + 6 x}{{(- x^{2} + 1)}^{4}}$

Step 2.1.2.5.4.1.3

Factor $2 x$ out of $6 x$ .

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

Step 2.1.2.5.4.1.4

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

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

Step 2.1.2.5.4.1.5

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

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

Step 2.1.2.5.4.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 2.1.2.5.4.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

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

Step 2.1.2.5.4.4

Factor by grouping.

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Step 2.1.2.5.4.4.1

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 $a \cdot c = - 1 \cdot 3 = - 3$ and whose sum is $b = - 2$ .

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Step 2.1.2.5.4.4.1.1

Factor $- 2$ out of $- 2 u$ .

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

Step 2.1.2.5.4.4.1.2

Rewrite $- 2$ as $1$ plus $- 3$

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

Step 2.1.2.5.4.4.1.3

Apply the distributive property.

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

Step 2.1.2.5.4.4.1.4

Multiply $u$ by $1$ .

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

Step 2.1.2.5.4.4.2

Factor out the greatest common factor from each group.

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Step 2.1.2.5.4.4.2.1

Group the first two terms and the last two terms.

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

Step 2.1.2.5.4.4.2.2

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

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

Step 2.1.2.5.4.4.3

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

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

Step 2.1.2.5.4.5

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 2.1.2.5.4.6

Rewrite $1$ as $1^{2}$ .

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

Step 2.1.2.5.4.7

Reorder $- x^{2}$ and $1^{2}$ .

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

Step 2.1.2.5.4.8

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

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

Step 2.1.2.5.5

Simplify the denominator.

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Step 2.1.2.5.5.1

Rewrite $1$ as $1^{2}$ .

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

Step 2.1.2.5.5.2

Reorder $- x^{2}$ and $1^{2}$ .

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

Step 2.1.2.5.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

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

Step 2.1.2.5.5.4

Apply the product rule to $(1 + x) (1 - x)$ .

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

Step 2.1.2.5.6

Cancel the common factor of $1 + x$ and ${(1 + x)}^{4}$ .

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Step 2.1.2.5.6.1

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

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

Step 2.1.2.5.6.2

Cancel the common factors.

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Step 2.1.2.5.6.2.1

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

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

Step 2.1.2.5.6.2.2

Cancel the common factor.

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

Step 2.1.2.5.6.2.3

Rewrite the expression.

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

Step 2.1.2.5.7

Cancel the common factor of $1 - x$ and ${(1 - x)}^{4}$ .

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Step 2.1.2.5.7.1

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

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

Step 2.1.2.5.7.2

Cancel the common factors.

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Step 2.1.2.5.7.2.1

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

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

Step 2.1.2.5.7.2.2

Cancel the common factor.

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

Step 2.1.2.5.7.2.3

Rewrite the expression.

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

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

Step 2.1.3

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

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

Step 2.2

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

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Step 2.2.1

Set the second derivative equal to $0$ .

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

Step 2.2.2

Set the numerator equal to zero.

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

Step 2.2.3

Solve the equation for $x$ .

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Step 2.2.3.1

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

$x = 0$

$x^{2} + 3 = 0$

Step 2.2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.3.3

Set $x^{2} + 3$ equal to $0$ and solve for $x$ .

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Step 2.2.3.3.1

Set $x^{2} + 3$ equal to $0$ .

$x^{2} + 3 = 0$

Step 2.2.3.3.2

Solve $x^{2} + 3 = 0$ for $x$ .

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Step 2.2.3.3.2.1

Subtract $3$ from both sides of the equation.

$x^{2} = - 3$

Step 2.2.3.3.2.2

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

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

Step 2.2.3.3.2.3

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

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Step 2.2.3.3.2.3.1

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

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

Step 2.2.3.3.2.3.2

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

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

Step 2.2.3.3.2.3.3

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

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

Step 2.2.3.3.2.4

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

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Step 2.2.3.3.2.4.1

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

$x = i \sqrt{3}$

Step 2.2.3.3.2.4.2

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

$x = - i \sqrt{3}$

Step 2.2.3.3.2.4.3

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

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

Step 2.2.3.4

The final solution is all the values that make $2 x (x^{2} + 3) = 0$ true.

$x = 0, i \sqrt{3}, - i \sqrt{3}$

Step 3

Find the domain of $f (x) = \frac{x}{1 - x^{2}}$ .

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Step 3.1

Set the denominator in $\frac{x}{1 - x^{2}}$ equal to $0$ to find where the expression is undefined.

$1 - x^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$- x^{2} = - 1$

Step 3.2.2

Divide each term in $- x^{2} = - 1$ by $- 1$ and simplify.

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Step 3.2.2.1

Divide each term in $- x^{2} = - 1$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 1}{- 1}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 1}{- 1}$

Step 3.2.2.2.2

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

$x^{2} = \frac{- 1}{- 1}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $- 1$ by $- 1$ .

$x^{2} = 1$

Step 3.2.3

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

$x = \pm \sqrt{1}$

Step 3.2.4

Any root of $1$ is $1$ .

$x = \pm 1$

Step 3.2.5

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

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Step 3.2.5.1

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

$x = 1$

Step 3.2.5.2

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

$x = - 1$

Step 3.2.5.3

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

$x = 1, - 1$

Step 3.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 1) \cup (- 1, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x \neq 1, - 1}$

Interval Notation:

$(- \infty, - 1) \cup (- 1, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x \neq 1, - 1}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - 1) \cup (- 1, 0) \cup (0, 1) \cup (1, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - 1)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the expression.

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Step 5.2.1.1

Remove parentheses.

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

Step 5.2.1.2

Multiply $2$ by $- 4$ .

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

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Subtract $4$ from $1$ .

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

Step 5.2.2.2

Multiply $- 1$ by $- 4$ .

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

Step 5.2.2.3

Add $1$ and $4$ .

$f'' (- 4) = \frac{- 8 ({(- 4)}^{2} + 3)}{{(- 3)}^{3} \cdot 5^{3}}$

Step 5.2.2.4

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

$f'' (- 4) = \frac{- 8 ({(- 4)}^{2} + 3)}{- 27 \cdot 5^{3}}$

Step 5.2.2.5

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

$f'' (- 4) = \frac{- 8 ({(- 4)}^{2} + 3)}{- 27 \cdot 125}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

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

$f'' (- 4) = \frac{- 8 (16 + 3)}{- 27 \cdot 125}$

Step 5.2.3.2

Add $16$ and $3$ .

$f'' (- 4) = \frac{- 8 \cdot 19}{- 27 \cdot 125}$

Step 5.2.4

Reduce the expression by cancelling the common factors.

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Step 5.2.4.1

Multiply $- 27$ by $125$ .

$f'' (- 4) = \frac{- 8 \cdot 19}{- 3375}$

Step 5.2.4.2

Multiply $- 8$ by $19$ .

$f'' (- 4) = \frac{- 152}{- 3375}$

Step 5.2.4.3

Dividing two negative values results in a positive value.

$f'' (- 4) = \frac{152}{3375}$

Step 5.2.5

The final answer is $\frac{152}{3375}$ .

$\frac{152}{3375}$

Step 5.3

The graph is concave up on the interval $(- \infty, - 1)$ because $f'' (- 4)$ is positive.

Concave up on $(- \infty, - 1)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(- 1, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $- 0.5$ in the expression.

$f'' (- 0.5) = \frac{2 (- 0.5) ({(- 0.5)}^{2} + 3)}{{(1 - 0.5)}^{3} {(1 - (- 0.5))}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the expression.

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Step 6.2.1.1

Remove parentheses.

$f'' (- 0.5) = \frac{2 (- 0.5) ({(- 0.5)}^{2} + 3)}{{(1 - 0.5)}^{3} {(1 - (- 0.5))}^{3}}$

Step 6.2.1.2

Multiply $2$ by $- 0.5$ .

$f'' (- 0.5) = \frac{- 1 ({(- 0.5)}^{2} + 3)}{{(1 - 0.5)}^{3} {(1 - (- 0.5))}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Subtract $0.5$ from $1$ .

$f'' (- 0.5) = \frac{- 1 ({(- 0.5)}^{2} + 3)}{{0.5}^{3} {(1 - (- 0.5))}^{3}}$

Step 6.2.2.2

Multiply $- 1$ by $- 0.5$ .

$f'' (- 0.5) = \frac{- 1 ({(- 0.5)}^{2} + 3)}{{0.5}^{3} {(1 + 0.5)}^{3}}$

Step 6.2.2.3

Add $1$ and $0.5$ .

$f'' (- 0.5) = \frac{- 1 ({(- 0.5)}^{2} + 3)}{{0.5}^{3} \cdot {1.5}^{3}}$

Step 6.2.2.4

Raise $0.5$ to the power of $3$ .

$f'' (- 0.5) = \frac{- 1 ({(- 0.5)}^{2} + 3)}{0.125 \cdot {1.5}^{3}}$

Step 6.2.2.5

Raise $1.5$ to the power of $3$ .

$f'' (- 0.5) = \frac{- 1 ({(- 0.5)}^{2} + 3)}{0.125 \cdot 3.375}$

Step 6.2.3

Simplify the numerator.

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Step 6.2.3.1

Raise $- 0.5$ to the power of $2$ .

$f'' (- 0.5) = \frac{- 1 (0.25 + 3)}{0.125 \cdot 3.375}$

Step 6.2.3.2

Add $0.25$ and $3$ .

$f'' (- 0.5) = \frac{- 1 \cdot 3.25}{0.125 \cdot 3.375}$

Step 6.2.4

Simplify the expression.

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Step 6.2.4.1

Multiply $0.125$ by $3.375$ .

$f'' (- 0.5) = \frac{- 1 \cdot 3.25}{0.421875}$

Step 6.2.4.2

Multiply $- 1$ by $3.25$ .

$f'' (- 0.5) = \frac{- 3.25}{0.421875}$

Step 6.2.4.3

Divide $- 3.25$ by $0.421875$ .

$f'' (- 0.5) = - 7.\overline{703}$

Step 6.2.5

The final answer is $- 7.\overline{703}$ .

$- 7.\overline{703}$

Step 6.3

The graph is concave down on the interval $(- 1, 0)$ because $f'' (- 0.5)$ is negative.

Concave down on $(- 1, 0)$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(0, 1)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $0.5$ in the expression.

$f'' (0.5) = \frac{2 (0.5) ({(0.5)}^{2} + 3)}{{(1 + 0.5)}^{3} {(1 - (0.5))}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Remove parentheses.

$f'' (0.5) = \frac{2 (0.5) ({(0.5)}^{2} + 3)}{{(1 + 0.5)}^{3} {(1 - (0.5))}^{3}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Multiply $2$ by $0.5$ .

$f'' (0.5) = \frac{1 ({0.5}^{2} + 3)}{{(1 + 0.5)}^{3} {(1 - (0.5))}^{3}}$

Step 7.2.2.2

Multiply ${0.5}^{2} + 3$ by $1$ .

$f'' (0.5) = \frac{{0.5}^{2} + 3}{{(1 + 0.5)}^{3} {(1 - (0.5))}^{3}}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

Add $1$ and $0.5$ .

$f'' (0.5) = \frac{{0.5}^{2} + 3}{{1.5}^{3} {(1 - (0.5))}^{3}}$

Step 7.2.3.2

Multiply $- 1$ by $0.5$ .

$f'' (0.5) = \frac{{0.5}^{2} + 3}{{1.5}^{3} {(1 - 0.5)}^{3}}$

Step 7.2.3.3

Subtract $0.5$ from $1$ .

$f'' (0.5) = \frac{{0.5}^{2} + 3}{{1.5}^{3} \cdot {0.5}^{3}}$

Step 7.2.3.4

Raise $1.5$ to the power of $3$ .

$f'' (0.5) = \frac{{0.5}^{2} + 3}{3.375 \cdot {0.5}^{3}}$

Step 7.2.3.5

Raise $0.5$ to the power of $3$ .

$f'' (0.5) = \frac{{0.5}^{2} + 3}{3.375 \cdot 0.125}$

Step 7.2.4

Simplify the numerator.

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Step 7.2.4.1

Raise $0.5$ to the power of $2$ .

$f'' (0.5) = \frac{0.25 + 3}{3.375 \cdot 0.125}$

Step 7.2.4.2

Add $0.25$ and $3$ .

$f'' (0.5) = \frac{3.25}{3.375 \cdot 0.125}$

Step 7.2.5

Simplify the expression.

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Step 7.2.5.1

Multiply $3.375$ by $0.125$ .

$f'' (0.5) = \frac{3.25}{0.421875}$

Step 7.2.5.2

Divide $3.25$ by $0.421875$ .

$f'' (0.5) = 7.\overline{703}$

Step 7.2.6

The final answer is $7.\overline{703}$ .

$7.\overline{703}$

Step 7.3

The graph is concave up on the interval $(0, 1)$ because $f'' (0.5)$ is positive.

Concave up on $(0, 1)$ since $f'' (x)$ is positive

Step 8

Substitute any number from the interval $(1, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 8.1

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

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

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the expression.

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Step 8.2.1.1

Remove parentheses.

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

Step 8.2.1.2

Multiply $2$ by $4$ .

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

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Add $1$ and $4$ .

$f'' (4) = \frac{8 (4^{2} + 3)}{5^{3} {(1 - (4))}^{3}}$

Step 8.2.2.2

Multiply $- 1$ by $4$ .

$f'' (4) = \frac{8 (4^{2} + 3)}{5^{3} {(1 - 4)}^{3}}$

Step 8.2.2.3

Subtract $4$ from $1$ .

$f'' (4) = \frac{8 (4^{2} + 3)}{5^{3} {(- 3)}^{3}}$

Step 8.2.2.4

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

$f'' (4) = \frac{8 (4^{2} + 3)}{125 {(- 3)}^{3}}$

Step 8.2.2.5

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

$f'' (4) = \frac{8 (4^{2} + 3)}{125 \cdot - 27}$

Step 8.2.3

Simplify the numerator.

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Step 8.2.3.1

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

$f'' (4) = \frac{8 (16 + 3)}{125 \cdot - 27}$

Step 8.2.3.2

Add $16$ and $3$ .

$f'' (4) = \frac{8 \cdot 19}{125 \cdot - 27}$

Step 8.2.4

Simplify the expression.

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Step 8.2.4.1

Multiply $125$ by $- 27$ .

$f'' (4) = \frac{8 \cdot 19}{- 3375}$

Step 8.2.4.2

Multiply $8$ by $19$ .

$f'' (4) = \frac{152}{- 3375}$

Step 8.2.4.3

Move the negative in front of the fraction.

$f'' (4) = - \frac{152}{3375}$

Step 8.2.5

The final answer is $- \frac{152}{3375}$ .

$- \frac{152}{3375}$

Step 8.3

The graph is concave down on the interval $(1, \infty)$ because $f'' (4)$ is negative.

Concave down on $(1, \infty)$ since $f'' (x)$ is negative

Step 9

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, - 1)$ since $f'' (x)$ is positive

Concave down on $(- 1, 0)$ since $f'' (x)$ is negative

Concave up on $(0, 1)$ since $f'' (x)$ is positive

Concave down on $(1, \infty)$ since $f'' (x)$ is negative

Step 10