Calculus Examples

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

Step 1

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

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

Step 2

Find the first derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Add $1$ and $0$ .

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

Step 2.3.5.2

Multiply $2$ by $1$ .

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

Step 2.3.6

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

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

Step 2.3.7

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

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

Step 2.3.8

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

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

Step 2.3.9

Simplify the expression.

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Step 2.3.9.1

Add $2 x$ and $0$ .

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

Step 2.3.9.2

Multiply $2$ by $- 1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Simplify the numerator.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Multiply $2$ by $1$ .

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

Step 2.4.2.1.2

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

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Step 2.4.2.1.2.1

Apply the distributive property.

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

Step 2.4.2.1.2.2

Apply the distributive property.

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

Step 2.4.2.1.2.3

Apply the distributive property.

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

Step 2.4.2.1.3

Simplify each term.

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Step 2.4.2.1.3.1

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

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Step 2.4.2.1.3.1.1

Move $x$ .

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

Step 2.4.2.1.3.1.2

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

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Step 2.4.2.1.3.1.2.1

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

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

Step 2.4.2.1.3.1.2.2

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

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

Step 2.4.2.1.3.1.3

Add $1$ and $2$ .

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

Step 2.4.2.1.3.2

Multiply $- 1$ by $2$ .

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

Step 2.4.2.1.3.3

Multiply $2$ by $- 1$ .

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

Step 2.4.2.1.4

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

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

Step 2.4.2.1.5

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

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Step 2.4.2.1.5.1

Apply the distributive property.

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

Step 2.4.2.1.5.2

Apply the distributive property.

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

Step 2.4.2.1.5.3

Apply the distributive property.

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

Step 2.4.2.1.6

Simplify and combine like terms.

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Step 2.4.2.1.6.1

Simplify each term.

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Step 2.4.2.1.6.1.1

Multiply $x$ by $x$ .

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

Step 2.4.2.1.6.1.2

Move $- 1$ to the left of $x$ .

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

Step 2.4.2.1.6.1.3

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

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

Step 2.4.2.1.6.1.4

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

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

Step 2.4.2.1.6.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.4.2.1.6.2

Subtract $x$ from $- x$ .

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

Step 2.4.2.1.7

Apply the distributive property.

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

Step 2.4.2.1.8

Simplify.

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Step 2.4.2.1.8.1

Multiply $- 2$ by $- 2$ .

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

Step 2.4.2.1.8.2

Multiply $- 2$ by $1$ .

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

Step 2.4.2.1.9

Apply the distributive property.

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

Step 2.4.2.1.10

Simplify.

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Step 2.4.2.1.10.1

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

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Step 2.4.2.1.10.1.1

Move $x$ .

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

Step 2.4.2.1.10.1.2

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

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Step 2.4.2.1.10.1.2.1

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

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

Step 2.4.2.1.10.1.2.2

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

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

Step 2.4.2.1.10.1.3

Add $1$ and $2$ .

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

Step 2.4.2.1.10.2

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

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Step 2.4.2.1.10.2.1

Move $x$ .

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

Step 2.4.2.1.10.2.2

Multiply $x$ by $x$ .

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

Step 2.4.2.2

Combine the opposite terms in $2 x^{3} - 2 x^{2} + 2 x - 2 - 2 x^{3} + 4 x^{2} - 2 x$ .

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Step 2.4.2.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

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

Step 2.4.2.2.2

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

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

Step 2.4.2.2.3

Subtract $2 x$ from $2 x$ .

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

Step 2.4.2.2.4

Add $- 2 x^{2} - 2 + 4 x^{2}$ and $0$ .

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

Step 2.4.2.3

Add $- 2 x^{2}$ and $4 x^{2}$ .

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

Step 2.4.3

Simplify the numerator.

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Step 2.4.3.1

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

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Step 2.4.3.1.1

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

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

Step 2.4.3.1.2

Factor $2$ out of $- 2$ .

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

Step 2.4.3.1.3

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

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

Step 2.4.3.2

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

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

Step 2.4.3.3

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 $b = 1$ .

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

Step 3

Find the second derivative of the function.

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Step 3.1

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

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

Step 3.2

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $2$ by $2$ .

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

Step 3.4

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 $f (x) = x + 1$ and $g (x) = x - 1$ .

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

Step 3.5

Differentiate.

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

$f'' (x) = 2 (\frac{{(x^{2} + 1)}^{2} ((x + 1) (1 + 0) + (x - 1) \frac{d}{d x} (x + 1)) - (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{2}}{{(x^{2} + 1)}^{4}})$

Step 3.5.4

Simplify the expression.

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Step 3.5.4.1

Add $1$ and $0$ .

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

Step 3.5.4.2

Multiply $x + 1$ by $1$ .

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

Step 3.5.5

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

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

Step 3.5.6

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

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

Step 3.5.7

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

$f'' (x) = 2 (\frac{{(x^{2} + 1)}^{2} (x + 1 + (x - 1) (1 + 0)) - (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{2}}{{(x^{2} + 1)}^{4}})$

Step 3.5.8

Simplify by adding terms.

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Step 3.5.8.1

Add $1$ and $0$ .

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

Step 3.5.8.2

Multiply $x - 1$ by $1$ .

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

Step 3.5.8.3

Add $x$ and $x$ .

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

Step 3.5.8.4

Simplify the expression.

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Step 3.5.8.4.1

Subtract $1$ from $1$ .

$f'' (x) = 2 (\frac{{(x^{2} + 1)}^{2} (2 x + 0) - (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{2}}{{(x^{2} + 1)}^{4}})$

Step 3.5.8.4.2

Add $2 x$ and $0$ .

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

Step 3.5.8.4.3

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

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

Step 3.6

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 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.7

Differentiate.

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Step 3.7.1

Multiply $2$ by $- 1$ .

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

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

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

Step 3.7.5

Combine fractions.

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Step 3.7.5.1

Add $2 x$ and $0$ .

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

Step 3.7.5.2

Simplify the expression.

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Step 3.7.5.2.1

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

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

Step 3.7.5.2.2

Multiply $2$ by $- 2$ .

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

Step 3.7.5.3

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

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

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Apply the distributive property.

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

Step 3.8.3

Apply the distributive property.

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

Step 3.8.4

Simplify the numerator.

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Step 3.8.4.1

Simplify each term.

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Step 3.8.4.1.1

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

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

Step 3.8.4.1.2

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

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Step 3.8.4.1.2.1

Apply the distributive property.

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

Step 3.8.4.1.2.2

Apply the distributive property.

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

Step 3.8.4.1.2.3

Apply the distributive property.

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

Step 3.8.4.1.3

Simplify and combine like terms.

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Step 3.8.4.1.3.1

Simplify each term.

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Step 3.8.4.1.3.1.1

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

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Step 3.8.4.1.3.1.1.1

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

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

Step 3.8.4.1.3.1.1.2

Add $2$ and $2$ .

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

Step 3.8.4.1.3.1.2

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

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

Step 3.8.4.1.3.1.3

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

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

Step 3.8.4.1.3.1.4

Multiply $1$ by $1$ .

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

Step 3.8.4.1.3.2

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

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

Step 3.8.4.1.4

Apply the distributive property.

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

Step 3.8.4.1.5

Simplify.

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Step 3.8.4.1.5.1

Multiply $2$ by $2$ .

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

Step 3.8.4.1.5.2

Multiply $2$ by $1$ .

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

Step 3.8.4.1.6

Apply the distributive property.

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

Step 3.8.4.1.7

Simplify.

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Step 3.8.4.1.7.1

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

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Step 3.8.4.1.7.1.1

Move $x$ .

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

Step 3.8.4.1.7.1.2

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

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Step 3.8.4.1.7.1.2.1

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

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

Step 3.8.4.1.7.1.2.2

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

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

Step 3.8.4.1.7.1.3

Add $1$ and $4$ .

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

Step 3.8.4.1.7.2

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

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Step 3.8.4.1.7.2.1

Move $x$ .

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

Step 3.8.4.1.7.2.2

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

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Step 3.8.4.1.7.2.2.1

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

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

Step 3.8.4.1.7.2.2.2

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

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

Step 3.8.4.1.7.2.3

Add $1$ and $2$ .

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

Step 3.8.4.1.8

Apply the distributive property.

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

Step 3.8.4.1.9

Simplify.

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Step 3.8.4.1.9.1

Multiply $2$ by $2$ .

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

Step 3.8.4.1.9.2

Multiply $4$ by $2$ .

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

Step 3.8.4.1.9.3

Multiply $2$ by $2$ .

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

Step 3.8.4.1.10

Multiply $- 4$ by $1$ .

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

Step 3.8.4.1.11

Expand $(- 4 x - 4) (x - 1)$ using the FOIL Method.

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Step 3.8.4.1.11.1

Apply the distributive property.

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

Step 3.8.4.1.11.2

Apply the distributive property.

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

Step 3.8.4.1.11.3

Apply the distributive property.

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

Step 3.8.4.1.12

Simplify and combine like terms.

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Step 3.8.4.1.12.1

Simplify each term.

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Step 3.8.4.1.12.1.1

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

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Step 3.8.4.1.12.1.1.1

Move $x$ .

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

Step 3.8.4.1.12.1.1.2

Multiply $x$ by $x$ .

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

Step 3.8.4.1.12.1.2

Multiply $- 1$ by $- 4$ .

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

Step 3.8.4.1.12.1.3

Multiply $- 4$ by $- 1$ .

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

Step 3.8.4.1.12.2

Subtract $4 x$ from $4 x$ .

$f'' (x) = \frac{4 x^{5} + 8 x^{3} + 4 x + 2 ((- 4 x^{2} + 0 + 4) (x^{2} x + 1 x))}{{(x^{2} + 1)}^{4}}$

Step 3.8.4.1.12.3

Add $- 4 x^{2}$ and $0$ .

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

Step 3.8.4.1.13

Simplify each term.

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Step 3.8.4.1.13.1

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

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Step 3.8.4.1.13.1.1

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

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Step 3.8.4.1.13.1.1.1

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

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

Step 3.8.4.1.13.1.1.2

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

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

Step 3.8.4.1.13.1.2

Add $2$ and $1$ .

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

Step 3.8.4.1.13.2

Multiply $x$ by $1$ .

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

Step 3.8.4.1.14

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

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Step 3.8.4.1.14.1

Apply the distributive property.

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

Step 3.8.4.1.14.2

Apply the distributive property.

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

Step 3.8.4.1.14.3

Apply the distributive property.

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

Step 3.8.4.1.15

Simplify and combine like terms.

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Step 3.8.4.1.15.1

Simplify each term.

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Step 3.8.4.1.15.1.1

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

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Step 3.8.4.1.15.1.1.1

Move $x^{3}$ .

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

Step 3.8.4.1.15.1.1.2

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

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

Step 3.8.4.1.15.1.1.3

Add $3$ and $2$ .

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

Step 3.8.4.1.15.1.2

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

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Step 3.8.4.1.15.1.2.1

Move $x$ .

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

Step 3.8.4.1.15.1.2.2

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

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Step 3.8.4.1.15.1.2.2.1

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

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

Step 3.8.4.1.15.1.2.2.2

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

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

Step 3.8.4.1.15.1.2.3

Add $1$ and $2$ .

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

Step 3.8.4.1.15.2

Add $- 4 x^{3}$ and $4 x^{3}$ .

$f'' (x) = \frac{4 x^{5} + 8 x^{3} + 4 x + 2 (- 4 x^{5} + 0 + 4 x)}{{(x^{2} + 1)}^{4}}$

Step 3.8.4.1.15.3

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

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

Step 3.8.4.1.16

Apply the distributive property.

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

Step 3.8.4.1.17

Multiply $- 4$ by $2$ .

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

Step 3.8.4.1.18

Multiply $4$ by $2$ .

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

Step 3.8.4.2

Subtract $8 x^{5}$ from $4 x^{5}$ .

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

Step 3.8.4.3

Add $4 x$ and $8 x$ .

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

Step 3.8.5

Simplify the numerator.

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Step 3.8.5.1

Factor $4 x$ out of $- 4 x^{5} + 8 x^{3} + 12 x$ .

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Step 3.8.5.1.1

Factor $4 x$ out of $- 4 x^{5}$ .

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

Step 3.8.5.1.2

Factor $4 x$ out of $8 x^{3}$ .

$f'' (x) = \frac{4 x (- x^{4}) + 4 x (2 x^{2}) + 12 x}{{(x^{2} + 1)}^{4}}$

Step 3.8.5.1.3

Factor $4 x$ out of $12 x$ .

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

Step 3.8.5.1.4

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

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

Step 3.8.5.1.5

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

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

Step 3.8.5.2

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

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

Step 3.8.5.3

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

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

Step 3.8.5.4

Factor by grouping.

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Step 3.8.5.4.1

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 3.8.5.4.1.1

Factor $2$ out of $2 u$ .

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

Step 3.8.5.4.1.2

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

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

Step 3.8.5.4.1.3

Apply the distributive property.

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

Step 3.8.5.4.2

Factor out the greatest common factor from each group.

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Step 3.8.5.4.2.1

Group the first two terms and the last two terms.

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

Step 3.8.5.4.2.2

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

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

Step 3.8.5.4.3

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

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

Step 3.8.5.5

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

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

Step 3.8.6

Cancel the common factor of $- x^{2} - 1$ and ${(x^{2} + 1)}^{4}$ .

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Step 3.8.6.1

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

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

Step 3.8.6.2

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

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

Step 3.8.6.3

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

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

Step 3.8.6.4

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

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

Step 3.8.6.5

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

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

Step 3.8.6.6

Cancel the common factors.

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Step 3.8.6.6.1

Factor $x^{2} + 1$ out of ${(x^{2} + 1)}^{4}$ .

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

Step 3.8.6.6.2

Cancel the common factor.

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

Step 3.8.6.6.3

Rewrite the expression.

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

Step 3.8.7

Multiply $- 1$ by $4$ .

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

Step 3.8.8

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

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

Step 5.1.2

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

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Step 5.1.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 5.1.2.2

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

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

Step 5.1.2.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 5.1.3

Differentiate.

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Step 5.1.3.1

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

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

Step 5.1.3.2

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

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

Step 5.1.3.3

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

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

Step 5.1.3.4

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

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

Step 5.1.3.5

Simplify the expression.

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Step 5.1.3.5.1

Add $1$ and $0$ .

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

Step 5.1.3.5.2

Multiply $2$ by $1$ .

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

Step 5.1.3.6

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

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

Step 5.1.3.7

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

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

Step 5.1.3.8

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

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

Step 5.1.3.9

Simplify the expression.

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Step 5.1.3.9.1

Add $2 x$ and $0$ .

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

Step 5.1.3.9.2

Multiply $2$ by $- 1$ .

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

Step 5.1.4

Simplify.

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Step 5.1.4.1

Apply the distributive property.

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

Step 5.1.4.2

Simplify the numerator.

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Step 5.1.4.2.1

Simplify each term.

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Step 5.1.4.2.1.1

Multiply $2$ by $1$ .

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

Step 5.1.4.2.1.2

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

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Step 5.1.4.2.1.2.1

Apply the distributive property.

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

Step 5.1.4.2.1.2.2

Apply the distributive property.

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

Step 5.1.4.2.1.2.3

Apply the distributive property.

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

Step 5.1.4.2.1.3

Simplify each term.

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Step 5.1.4.2.1.3.1

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

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Step 5.1.4.2.1.3.1.1

Move $x$ .

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

Step 5.1.4.2.1.3.1.2

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

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Step 5.1.4.2.1.3.1.2.1

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

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

Step 5.1.4.2.1.3.1.2.2

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

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

Step 5.1.4.2.1.3.1.3

Add $1$ and $2$ .

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

Step 5.1.4.2.1.3.2

Multiply $- 1$ by $2$ .

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

Step 5.1.4.2.1.3.3

Multiply $2$ by $- 1$ .

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

Step 5.1.4.2.1.4

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

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

Step 5.1.4.2.1.5

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

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Step 5.1.4.2.1.5.1

Apply the distributive property.

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

Step 5.1.4.2.1.5.2

Apply the distributive property.

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

Step 5.1.4.2.1.5.3

Apply the distributive property.

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

Step 5.1.4.2.1.6

Simplify and combine like terms.

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Step 5.1.4.2.1.6.1

Simplify each term.

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Step 5.1.4.2.1.6.1.1

Multiply $x$ by $x$ .

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

Step 5.1.4.2.1.6.1.2

Move $- 1$ to the left of $x$ .

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

Step 5.1.4.2.1.6.1.3

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

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

Step 5.1.4.2.1.6.1.4

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

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

Step 5.1.4.2.1.6.1.5

Multiply $- 1$ by $- 1$ .

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

Step 5.1.4.2.1.6.2

Subtract $x$ from $- x$ .

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

Step 5.1.4.2.1.7

Apply the distributive property.

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

Step 5.1.4.2.1.8

Simplify.

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Step 5.1.4.2.1.8.1

Multiply $- 2$ by $- 2$ .

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

Step 5.1.4.2.1.8.2

Multiply $- 2$ by $1$ .

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

Step 5.1.4.2.1.9

Apply the distributive property.

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

Step 5.1.4.2.1.10

Simplify.

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Step 5.1.4.2.1.10.1

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

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Step 5.1.4.2.1.10.1.1

Move $x$ .

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

Step 5.1.4.2.1.10.1.2

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

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Step 5.1.4.2.1.10.1.2.1

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

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

Step 5.1.4.2.1.10.1.2.2

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

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

Step 5.1.4.2.1.10.1.3

Add $1$ and $2$ .

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

Step 5.1.4.2.1.10.2

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

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Step 5.1.4.2.1.10.2.1

Move $x$ .

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

Step 5.1.4.2.1.10.2.2

Multiply $x$ by $x$ .

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

Step 5.1.4.2.2

Combine the opposite terms in $2 x^{3} - 2 x^{2} + 2 x - 2 - 2 x^{3} + 4 x^{2} - 2 x$ .

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Step 5.1.4.2.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

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

Step 5.1.4.2.2.2

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

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

Step 5.1.4.2.2.3

Subtract $2 x$ from $2 x$ .

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

Step 5.1.4.2.2.4

Add $- 2 x^{2} - 2 + 4 x^{2}$ and $0$ .

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

Step 5.1.4.2.3

Add $- 2 x^{2}$ and $4 x^{2}$ .

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

Step 5.1.4.3

Simplify the numerator.

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Step 5.1.4.3.1

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

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Step 5.1.4.3.1.1

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

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

Step 5.1.4.3.1.2

Factor $2$ out of $- 2$ .

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

Step 5.1.4.3.1.3

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

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

Step 5.1.4.3.2

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

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

Step 5.1.4.3.3

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 $b = 1$ .

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

Step 5.2

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

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

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Set the numerator equal to zero.

$2 (x + 1) (x - 1) = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

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

$x + 1 = 0$

$x - 1 = 0$

Step 6.3.2

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

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Step 6.3.2.1

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

$x + 1 = 0$

Step 6.3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 6.3.3

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

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Step 6.3.3.1

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

$x - 1 = 0$

Step 6.3.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 6.3.4

The final solution is all the values that make $2 (x + 1) (x - 1) = 0$ true.

$x = - 1, 1$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 8

Critical points to evaluate.

$x = - 1, 1$

Step 9

Evaluate the second derivative at $x = - 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 10

Evaluate the second derivative.

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Step 10.1

Multiply $4$ by $- 1$ .

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

Step 10.2

Simplify the denominator.

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Step 10.2.1

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

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

Step 10.2.2

Add $1$ and $1$ .

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

Step 10.2.3

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

$- \frac{- 4 ({(- 1)}^{2} - 3)}{8}$

Step 10.3

Simplify the numerator.

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Step 10.3.1

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

$- \frac{- 4 (1 - 3)}{8}$

Step 10.3.2

Subtract $3$ from $1$ .

$- \frac{- 4 \cdot - 2}{8}$

Step 10.4

Reduce the expression by cancelling the common factors.

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Step 10.4.1

Multiply $- 4$ by $- 2$ .

$- \frac{8}{8}$

Step 10.4.2

Cancel the common factor of $8$ .

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Step 10.4.2.1

Cancel the common factor.

$- \frac{8}{8}$

Step 10.4.2.2

Rewrite the expression.

$- 1 \cdot 1$

Step 10.4.3

Multiply $- 1$ by $1$ .

$- 1$

Step 11

$x = - 1$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 1$ is a local maximum

Step 12

Find the y-value when $x = - 1$ .

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Step 12.1

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

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

Step 12.2

Simplify the result.

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Step 12.2.1

Cancel the common factor of ${((- 1) - 1)}^{2}$ and ${(- 1)}^{2} + 1$ .

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Step 12.2.1.1

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

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

Step 12.2.1.2

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

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

Step 12.2.1.3

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

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

Step 12.2.1.4

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

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

Step 12.2.1.5

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

$f (- 1) = \frac{(- 1 - 1) (- 1 - 1)}{- 1 (- 1 - 1)}$

Step 12.2.1.6

Move the negative one from the denominator of $\frac{- 1 - 1}{- 1}$ .

$f (- 1) = - 1 \cdot (- 1 - 1)$

Step 12.2.2

Simplify the expression.

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Step 12.2.2.1

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

$f (- 1) = - (- 1 - 1)$

Step 12.2.2.2

Subtract $1$ from $- 1$ .

$f (- 1) = 2$

Step 12.2.2.3

Multiply $- 1$ by $- 2$ .

$f (- 1) = 2$

Step 12.2.3

The final answer is $2$ .

$y = 2$

Step 13

Evaluate the second derivative at $x = 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 14

Evaluate the second derivative.

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Step 14.1

Multiply $4$ by $1$ .

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

Step 14.2

Simplify the denominator.

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Step 14.2.1

One to any power is one.

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

Step 14.2.2

Add $1$ and $1$ .

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

Step 14.2.3

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

$- \frac{4 (1^{2} - 3)}{8}$

Step 14.3

Simplify the numerator.

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Step 14.3.1

One to any power is one.

$- \frac{4 (1 - 3)}{8}$

Step 14.3.2

Subtract $3$ from $1$ .

$- \frac{4 \cdot - 2}{8}$

Step 14.4

Simplify the expression.

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Step 14.4.1

Multiply $4$ by $- 2$ .

$- \frac{- 8}{8}$

Step 14.4.2

Divide $- 8$ by $8$ .

$- - 1$

Step 14.4.3

Multiply $- 1$ by $- 1$ .

$1$

Step 15

$x = 1$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 1$ is a local minimum

Step 16

Find the y-value when $x = 1$ .

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Step 16.1

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

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

Step 16.2

Simplify the result.

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Step 16.2.1

Simplify the numerator.

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Step 16.2.1.1

Subtract $1$ from $1$ .

$f (1) = \frac{0^{2}}{1^{2} + 1}$

Step 16.2.1.2

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

$f (1) = \frac{0}{1^{2} + 1}$

Step 16.2.2

Simplify the denominator.

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Step 16.2.2.1

One to any power is one.

$f (1) = \frac{0}{1 + 1}$

Step 16.2.2.2

Add $1$ and $1$ .

$f (1) = \frac{0}{2}$

Step 16.2.3

Divide $0$ by $2$ .

$f (1) = 0$

Step 16.2.4

The final answer is $0$ .

$y = 0$

Step 17

These are the local extrema for $f (x) = \frac{{(x - 1)}^{2}}{x^{2} + 1}$ .

$(- 1, 2)$ is a local maxima

$(1, 0)$ is a local minima

Step 18