Calculus Examples

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

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 x^{3}}{x^{4} + 1}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x^{3}}{x^{4} + 1}]$ .

$2 \frac{d}{d x} [\frac{x^{3}}{x^{4} + 1}]$

Step 1.1.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{3}$ and $g (x) = x^{4} + 1$ .

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

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

Step 1.1.3.2

Move $3$ to the left of $x^{4} + 1$ .

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

Step 1.1.3.3

By the Sum Rule, the derivative of $x^{4} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [1]$ .

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

Step 1.1.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Step 1.1.3.5

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

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

Step 1.1.3.6

Simplify the expression.

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Step 1.1.3.6.1

Add $4 x^{3}$ and $0$ .

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

Step 1.1.3.6.2

Multiply $4$ by $- 1$ .

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

Step 1.1.4

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

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Step 1.1.4.1

Move $x^{3}$ .

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

Step 1.1.4.2

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

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

Step 1.1.4.3

Add $3$ and $3$ .

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

Step 1.1.5

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

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

Step 1.1.6

Simplify.

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Step 1.1.6.1

Apply the distributive property.

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

Step 1.1.6.2

Apply the distributive property.

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

Step 1.1.6.3

Apply the distributive property.

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

Step 1.1.6.4

Simplify the numerator.

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Step 1.1.6.4.1

Simplify each term.

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Step 1.1.6.4.1.1

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 1.1.6.4.1.1.1

Move $x^{2}$ .

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

Step 1.1.6.4.1.1.2

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

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

Step 1.1.6.4.1.1.3

Add $2$ and $4$ .

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

Step 1.1.6.4.1.2

Multiply $3$ by $2$ .

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

Step 1.1.6.4.1.3

Multiply $3$ by $1$ .

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

Step 1.1.6.4.1.4

Multiply $3$ by $2$ .

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

Step 1.1.6.4.1.5

Multiply $- 4$ by $2$ .

$\frac{6 x^{6} + 6 x^{2} - 8 x^{6}}{{(x^{4} + 1)}^{2}}$

Step 1.1.6.4.2

Subtract $8 x^{6}$ from $6 x^{6}$ .

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

Step 1.1.6.5

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

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Step 1.1.6.5.1

Factor $2 x^{2}$ out of $- 2 x^{6}$ .

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

Step 1.1.6.5.2

Factor $2 x^{2}$ out of $6 x^{2}$ .

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

Step 1.1.6.5.3

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

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

Step 1.1.6.6

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

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

Step 1.1.6.7

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

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

Step 1.1.6.8

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

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

Step 1.1.6.9

Rewrite $- (x^{4} - 3)$ as $- 1 (x^{4} - 3)$ .

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

Step 1.1.6.10

Move the negative in front of the fraction.

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

Step 1.2

Find the second derivative.

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Step 1.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2 x^{2} (x^{4} - 3)}{{(x^{4} + 1)}^{2}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{x^{2} (x^{4} - 3)}{{(x^{4} + 1)}^{2}}]$ .

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

Step 1.2.2

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

Step 1.2.3

Multiply the exponents in ${({(x^{4} + 1)}^{2})}^{2}$ .

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Step 1.2.3.1

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

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

Step 1.2.3.2

Multiply $2$ by $2$ .

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

Step 1.2.4

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 $f (x) = x^{2}$ and $g (x) = x^{4} - 3$ .

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

Step 1.2.5

Differentiate.

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Step 1.2.5.1

By the Sum Rule, the derivative of $x^{4} - 3$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 3]$ .

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

Step 1.2.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Step 1.2.5.3

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

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

Step 1.2.5.4

Add $4 x^{3}$ and $0$ .

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

Step 1.2.6

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

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Step 1.2.6.1

Move $x^{3}$ .

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

Step 1.2.6.2

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

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

Step 1.2.6.3

Add $3$ and $2$ .

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

Step 1.2.7

Differentiate using the Power Rule.

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Step 1.2.7.1

Move $4$ to the left of $x^{5}$ .

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

Step 1.2.7.2

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

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

Step 1.2.7.3

Move $2$ to the left of $x^{4} - 3$ .

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

Step 1.2.8

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 $g (x) = x^{4} + 1$ .

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Step 1.2.8.1

To apply the Chain Rule, set $u$ as $x^{4} + 1$ .

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

Step 1.2.8.2

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

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

Step 1.2.8.3

Replace all occurrences of $u$ with $x^{4} + 1$ .

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

Step 1.2.9

Simplify with factoring out.

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Step 1.2.9.1

Multiply $2$ by $- 1$ .

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

Step 1.2.9.2

Factor $x^{4} + 1$ out of ${(x^{4} + 1)}^{2} (4 x^{5} + 2 (x^{4} - 3) x) - 2 x^{2} (x^{4} - 3) ((x^{4} + 1) \frac{d}{d x} [x^{4} + 1])$ .

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Step 1.2.9.2.1

Factor $x^{4} + 1$ out of ${(x^{4} + 1)}^{2} (4 x^{5} + 2 (x^{4} - 3) x)$ .

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

Step 1.2.9.2.2

Factor $x^{4} + 1$ out of $- 2 x^{2} (x^{4} - 3) ((x^{4} + 1) \frac{d}{d x} [x^{4} + 1])$ .

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

Step 1.2.9.2.3

Factor $x^{4} + 1$ out of $(x^{4} + 1) ((x^{4} + 1) (4 x^{5} + 2 (x^{4} - 3) x)) + (x^{4} + 1) (- 2 x^{2} (x^{4} - 3) (\frac{d}{d x} [x^{4} + 1]))$ .

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

Step 1.2.10

Cancel the common factors.

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Step 1.2.10.1

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

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

Step 1.2.10.2

Cancel the common factor.

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

Step 1.2.10.3

Rewrite the expression.

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

Step 1.2.11

By the Sum Rule, the derivative of $x^{4} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [1]$ .

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

Step 1.2.12

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Step 1.2.13

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

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

Step 1.2.14

Simplify the expression.

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Step 1.2.14.1

Add $4 x^{3}$ and $0$ .

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

Step 1.2.14.2

Multiply $4$ by $- 2$ .

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

Step 1.2.15

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

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Step 1.2.15.1

Move $x^{3}$ .

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

Step 1.2.15.2

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

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

Step 1.2.15.3

Add $3$ and $2$ .

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

Step 1.2.16

Combine $- 2$ and $\frac{(x^{4} + 1) (4 x^{5} + 2 (x^{4} - 3) x) - 8 x^{5} (x^{4} - 3)}{{(x^{4} + 1)}^{3}}$ .

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

Step 1.2.17

Move the negative in front of the fraction.

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

Step 1.2.18

Simplify.

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Step 1.2.18.1

Apply the distributive property.

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

Step 1.2.18.2

Apply the distributive property.

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

Step 1.2.18.3

Apply the distributive property.

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

Step 1.2.18.4

Apply the distributive property.

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

Step 1.2.18.5

Simplify the numerator.

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Step 1.2.18.5.1

Simplify each term.

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Step 1.2.18.5.1.1

Simplify each term.

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Step 1.2.18.5.1.1.1

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

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Step 1.2.18.5.1.1.1.1

Move $x$ .

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

Step 1.2.18.5.1.1.1.2

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

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Step 1.2.18.5.1.1.1.2.1

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

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

Step 1.2.18.5.1.1.1.2.2

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

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

Step 1.2.18.5.1.1.1.3

Add $1$ and $4$ .

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

Step 1.2.18.5.1.1.2

Multiply $2$ by $- 3$ .

$- \frac{2 ((x^{4} + 1) (4 x^{5} + 2 x^{5} - 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.2

Add $4 x^{5}$ and $2 x^{5}$ .

$- \frac{2 ((x^{4} + 1) (6 x^{5} - 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.3

Expand $(x^{4} + 1) (6 x^{5} - 6 x)$ using the FOIL Method.

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Step 1.2.18.5.1.3.1

Apply the distributive property.

$- \frac{2 (x^{4} (6 x^{5} - 6 x) + 1 (6 x^{5} - 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.3.2

Apply the distributive property.

$- \frac{2 (x^{4} (6 x^{5}) + x^{4} (- 6 x) + 1 (6 x^{5} - 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.3.3

Apply the distributive property.

$- \frac{2 (x^{4} (6 x^{5}) + x^{4} (- 6 x) + 1 (6 x^{5}) + 1 (- 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4

Simplify and combine like terms.

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Step 1.2.18.5.1.4.1

Simplify each term.

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Step 1.2.18.5.1.4.1.1

Rewrite using the commutative property of multiplication.

$- \frac{2 (6 x^{4} x^{5} + x^{4} (- 6 x) + 1 (6 x^{5}) + 1 (- 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4.1.2

Multiply $x^{4}$ by $x^{5}$ by adding the exponents.

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Step 1.2.18.5.1.4.1.2.1

Move $x^{5}$ .

$- \frac{2 (6 (x^{5} x^{4}) + x^{4} (- 6 x) + 1 (6 x^{5}) + 1 (- 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4.1.2.2

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

$- \frac{2 (6 x^{5 + 4} + x^{4} (- 6 x) + 1 (6 x^{5}) + 1 (- 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4.1.2.3

Add $5$ and $4$ .

$- \frac{2 (6 x^{9} + x^{4} (- 6 x) + 1 (6 x^{5}) + 1 (- 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4.1.3

Rewrite using the commutative property of multiplication.

$- \frac{2 (6 x^{9} - 6 x^{4} x + 1 (6 x^{5}) + 1 (- 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4.1.4

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

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Step 1.2.18.5.1.4.1.4.1

Move $x$ .

$- \frac{2 (6 x^{9} - 6 (x \cdot x^{4}) + 1 (6 x^{5}) + 1 (- 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4.1.4.2

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

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Step 1.2.18.5.1.4.1.4.2.1

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

$- \frac{2 (6 x^{9} - 6 (x^{1} x^{4}) + 1 (6 x^{5}) + 1 (- 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4.1.4.2.2

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

$- \frac{2 (6 x^{9} - 6 x^{1 + 4} + 1 (6 x^{5}) + 1 (- 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4.1.4.3

Add $1$ and $4$ .

$- \frac{2 (6 x^{9} - 6 x^{5} + 1 (6 x^{5}) + 1 (- 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4.1.5

Multiply $6 x^{5}$ by $1$ .

$- \frac{2 (6 x^{9} - 6 x^{5} + 6 x^{5} + 1 (- 6 x)) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4.1.6

Multiply $- 6 x$ by $1$ .

$- \frac{2 (6 x^{9} - 6 x^{5} + 6 x^{5} - 6 x) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4.2

Add $- 6 x^{5}$ and $6 x^{5}$ .

$- \frac{2 (6 x^{9} + 0 - 6 x) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.4.3

Add $6 x^{9}$ and $0$ .

$- \frac{2 (6 x^{9} - 6 x) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.5

Apply the distributive property.

$- \frac{2 (6 x^{9}) + 2 (- 6 x) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.6

Multiply $6$ by $2$ .

$- \frac{12 x^{9} + 2 (- 6 x) + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.7

Multiply $- 6$ by $2$ .

$- \frac{12 x^{9} - 12 x + 2 (- 8 x^{5} x^{4}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.8

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

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Step 1.2.18.5.1.8.1

Move $x^{4}$ .

$- \frac{12 x^{9} - 12 x + 2 (- 8 (x^{4} x^{5})) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.8.2

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

$- \frac{12 x^{9} - 12 x + 2 (- 8 x^{4 + 5}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.8.3

Add $4$ and $5$ .

$- \frac{12 x^{9} - 12 x + 2 (- 8 x^{9}) + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.9

Multiply $- 8$ by $2$ .

$- \frac{12 x^{9} - 12 x - 16 x^{9} + 2 (- 8 x^{5} \cdot - 3)}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.10

Multiply $- 3$ by $- 8$ .

$- \frac{12 x^{9} - 12 x - 16 x^{9} + 2 (24 x^{5})}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.1.11

Multiply $24$ by $2$ .

$- \frac{12 x^{9} - 12 x - 16 x^{9} + 48 x^{5}}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.5.2

Subtract $16 x^{9}$ from $12 x^{9}$ .

$- \frac{- 4 x^{9} - 12 x + 48 x^{5}}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.6

Simplify the numerator.

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Step 1.2.18.6.1

Factor $4 x$ out of $- 4 x^{9} - 12 x + 48 x^{5}$ .

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Step 1.2.18.6.1.1

Factor $4 x$ out of $- 4 x^{9}$ .

$- \frac{4 x (- x^{8}) - 12 x + 48 x^{5}}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.6.1.2

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

$- \frac{4 x (- x^{8}) + 4 x (- 3) + 48 x^{5}}{{(x^{4} + 1)}^{3}}$

Step 1.2.18.6.1.3

Factor $4 x$ out of $48 x^{5}$ .

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

Step 1.2.18.6.1.4

Factor $4 x$ out of $4 x (- x^{8}) + 4 x (- 3)$ .

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

Step 1.2.18.6.1.5

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

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

Step 1.2.18.6.2

Reorder terms.

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

Step 1.2.18.7

Factor $- 1$ out of $- x^{8}$ .

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

Step 1.2.18.8

Factor $- 1$ out of $12 x^{4}$ .

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

Step 1.2.18.9

Factor $- 1$ out of $- (x^{8}) - (- 12 x^{4})$ .

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

Step 1.2.18.10

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

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

Step 1.2.18.11

Factor $- 1$ out of $- (x^{8} - 12 x^{4}) - 1 (3)$ .

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

Step 1.2.18.12

Rewrite $- (x^{8} - 12 x^{4} + 3)$ as $- 1 (x^{8} - 12 x^{4} + 3)$ .

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

Step 1.2.18.13

Move the negative in front of the fraction.

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

Step 1.2.18.14

Multiply $- 1$ by $- 1$ .

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

Step 1.2.18.15

Multiply $\frac{(4 x) (x^{8} - 12 x^{4} + 3)}{{(x^{4} + 1)}^{3}}$ by $1$ .

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

Step 1.2.18.16

Reorder factors in $\frac{(4 x) (x^{8} - 12 x^{4} + 3)}{{(x^{4} + 1)}^{3}}$ .

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

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{4 x (x^{8} - 12 x^{4} + 3)}{{(x^{4} + 1)}^{3}}$ .

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

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{4 x (x^{8} - 12 x^{4} + 3)}{{(x^{4} + 1)}^{3}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$\frac{4 x (x^{8} - 12 x^{4} + 3)}{{(x^{4} + 1)}^{3}} = 0$

Step 2.2

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

$x \approx - 1.85122502, - 0.7109206, 0, 0.7109206, 1.85122502$

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $- 1.85122502$ in $f (x) = \frac{2 x^{3}}{x^{4} + 1}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $- 1.85122502$ in the expression.

$f (- 1.85122502) = \frac{2 {(- 1.85122502)}^{3}}{{(- 1.85122502)}^{4} + 1}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Raise $- 1.85122502$ to the power of $3$ .

$f (- 1.85122502) = \frac{2 \cdot - 6.34421126}{{(- 1.85122502)}^{4} + 1}$

Step 3.1.2.2

Simplify the denominator.

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Step 3.1.2.2.1

Raise $- 1.85122502$ to the power of $4$ .

$f (- 1.85122502) = \frac{2 \cdot - 6.34421126}{11.74456266 + 1}$

Step 3.1.2.2.2

Add $11.74456266$ and $1$ .

$f (- 1.85122502) = \frac{2 \cdot - 6.34421126}{12.74456266}$

Step 3.1.2.3

Simplify the expression.

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Step 3.1.2.3.1

Multiply $2$ by $- 6.34421126$ .

$f (- 1.85122502) = \frac{- 12.68842253}{12.74456266}$

Step 3.1.2.3.2

Divide $- 12.68842253$ by $12.74456266$ .

$f (- 1.85122502) = - 0.99559497$

Step 3.1.2.4

The final answer is $- 0.99559497$ .

$- 0.99559497$

Step 3.2

The point found by substituting $- 1.85122502$ in $f (x) = \frac{2 x^{3}}{x^{4} + 1}$ is $(- 1.85122502, - 0.99559497)$ . This point can be an inflection point.

$(- 1.85122502, - 0.99559497)$

Step 3.3

Substitute $- 0.7109206$ in $f (x) = \frac{2 x^{3}}{x^{4} + 1}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- 0.7109206$ in the expression.

$f (- 0.7109206) = \frac{2 {(- 0.7109206)}^{3}}{{(- 0.7109206)}^{4} + 1}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Raise $- 0.7109206$ to the power of $3$ .

$f (- 0.7109206) = \frac{2 \cdot - 0.35930503}{{(- 0.7109206)}^{4} + 1}$

Step 3.3.2.2

Simplify the denominator.

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Step 3.3.2.2.1

Raise $- 0.7109206$ to the power of $4$ .

$f (- 0.7109206) = \frac{2 \cdot - 0.35930503}{0.25543735 + 1}$

Step 3.3.2.2.2

Add $0.25543735$ and $1$ .

$f (- 0.7109206) = \frac{2 \cdot - 0.35930503}{1.25543735}$

Step 3.3.2.3

Simplify the expression.

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Step 3.3.2.3.1

Multiply $2$ by $- 0.35930503$ .

$f (- 0.7109206) = \frac{- 0.71861007}{1.25543735}$

Step 3.3.2.3.2

Divide $- 0.71861007$ by $1.25543735$ .

$f (- 0.7109206) = - 0.57239819$

Step 3.3.2.4

The final answer is $- 0.57239819$ .

$- 0.57239819$

Step 3.4

The point found by substituting $- 0.7109206$ in $f (x) = \frac{2 x^{3}}{x^{4} + 1}$ is $(- 0.7109206, - 0.57239819)$ . This point can be an inflection point.

$(- 0.7109206, - 0.57239819)$

Step 3.5

Substitute $0$ in $f (x) = \frac{2 x^{3}}{x^{4} + 1}$ to find the value of $y$ .

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Step 3.5.1

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

$f (0) = \frac{2 {(0)}^{3}}{{(0)}^{4} + 1}$

Step 3.5.2

Simplify the result.

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Step 3.5.2.1

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

$f (0) = \frac{2 \cdot 0}{0^{4} + 1}$

Step 3.5.2.2

Simplify the denominator.

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Step 3.5.2.2.1

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

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

Step 3.5.2.2.2

Add $0$ and $1$ .

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

Step 3.5.2.3

Simplify the expression.

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Step 3.5.2.3.1

Multiply $2$ by $0$ .

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

Step 3.5.2.3.2

Divide $0$ by $1$ .

$f (0) = 0$

Step 3.5.2.4

The final answer is $0$ .

$0$

Step 3.6

The point found by substituting $0$ in $f (x) = \frac{2 x^{3}}{x^{4} + 1}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 3.7

Substitute $0.7109206$ in $f (x) = \frac{2 x^{3}}{x^{4} + 1}$ to find the value of $y$ .

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Step 3.7.1

Replace the variable $x$ with $0.7109206$ in the expression.

$f (0.7109206) = \frac{2 {(0.7109206)}^{3}}{{(0.7109206)}^{4} + 1}$

Step 3.7.2

Simplify the result.

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Step 3.7.2.1

Raise $0.7109206$ to the power of $3$ .

$f (0.7109206) = \frac{2 \cdot 0.35930503}{{0.7109206}^{4} + 1}$

Step 3.7.2.2

Simplify the denominator.

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Step 3.7.2.2.1

Raise $0.7109206$ to the power of $4$ .

$f (0.7109206) = \frac{2 \cdot 0.35930503}{0.25543735 + 1}$

Step 3.7.2.2.2

Add $0.25543735$ and $1$ .

$f (0.7109206) = \frac{2 \cdot 0.35930503}{1.25543735}$

Step 3.7.2.3

Simplify the expression.

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Step 3.7.2.3.1

Multiply $2$ by $0.35930503$ .

$f (0.7109206) = \frac{0.71861007}{1.25543735}$

Step 3.7.2.3.2

Divide $0.71861007$ by $1.25543735$ .

$f (0.7109206) = 0.57239819$

Step 3.7.2.4

The final answer is $0.57239819$ .

$0.57239819$

Step 3.8

The point found by substituting $0.7109206$ in $f (x) = \frac{2 x^{3}}{x^{4} + 1}$ is $(0.7109206, 0.57239819)$ . This point can be an inflection point.

$(0.7109206, 0.57239819)$

Step 3.9

Substitute $1.85122502$ in $f (x) = \frac{2 x^{3}}{x^{4} + 1}$ to find the value of $y$ .

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Step 3.9.1

Replace the variable $x$ with $1.85122502$ in the expression.

$f (1.85122502) = \frac{2 {(1.85122502)}^{3}}{{(1.85122502)}^{4} + 1}$

Step 3.9.2

Simplify the result.

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Step 3.9.2.1

Raise $1.85122502$ to the power of $3$ .

$f (1.85122502) = \frac{2 \cdot 6.34421126}{{1.85122502}^{4} + 1}$

Step 3.9.2.2

Simplify the denominator.

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Step 3.9.2.2.1

Raise $1.85122502$ to the power of $4$ .

$f (1.85122502) = \frac{2 \cdot 6.34421126}{11.74456266 + 1}$

Step 3.9.2.2.2

Add $11.74456266$ and $1$ .

$f (1.85122502) = \frac{2 \cdot 6.34421126}{12.74456266}$

Step 3.9.2.3

Simplify the expression.

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Step 3.9.2.3.1

Multiply $2$ by $6.34421126$ .

$f (1.85122502) = \frac{12.68842253}{12.74456266}$

Step 3.9.2.3.2

Divide $12.68842253$ by $12.74456266$ .

$f (1.85122502) = 0.99559497$

Step 3.9.2.4

The final answer is $0.99559497$ .

$0.99559497$

Step 3.10

The point found by substituting $1.85122502$ in $f (x) = \frac{2 x^{3}}{x^{4} + 1}$ is $(1.85122502, 0.99559497)$ . This point can be an inflection point.

$(1.85122502, 0.99559497)$

Step 3.11

Determine the points that could be inflection points.

$(- 1.85122502, - 0.99559497), (- 0.7109206, - 0.57239819), (0, 0), (0.7109206, 0.57239819), (1.85122502, 0.99559497)$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - 1.85122502) \cup (- 1.85122502, - 0.7109206) \cup (- 0.7109206, 0) \cup (0, 0.7109206) \cup (0.7109206, 1.85122502) \cup (1.85122502, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 1.85122502)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 1.95122502$ in the expression.

$f'' (- 1.95122502) = \frac{4 (- 1.95122502) ({(- 1.95122502)}^{8} - 12 {(- 1.95122502)}^{4} + 3)}{{({(- 1.95122502)}^{4} + 1)}^{3}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Multiply $4$ by $- 1.95122502$ .

$f'' (- 1.95122502) = \frac{- 7.80490009 ({(- 1.95122502)}^{8} - 12 {(- 1.95122502)}^{4} + 3)}{{({(- 1.95122502)}^{4} + 1)}^{3}}$

Step 5.2.1.2

Multiply $- 7.80490009$ by ${(- 1.95122502)}^{8} - 12 {(- 1.95122502)}^{4} + 3$ .

$f'' (- 1.95122502) = \frac{- 305.72871822}{{({(- 1.95122502)}^{4} + 1)}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Raise $- 1.95122502$ to the power of $4$ .

$f'' (- 1.95122502) = \frac{- 305.72871822}{{(14.49537411 + 1)}^{3}}$

Step 5.2.2.2

Add $14.49537411$ and $1$ .

$f'' (- 1.95122502) = \frac{- 305.72871822}{{15.49537411}^{3}}$

Step 5.2.2.3

Raise $15.49537411$ to the power of $3$ .

$f'' (- 1.95122502) = \frac{- 305.72871822}{3720.54188867}$

Step 5.2.3

Divide $- 305.72871822$ by $3720.54188867$ .

$f'' (- 1.95122502) = - 0.08217316$

Step 5.2.4

The final answer is $- 0.08217316$ .

$- 0.08217316$

Step 5.3

At $- 1.95122502$ , the second derivative is $- 0.08217316$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 1.85122502)$

Decreasing on $(- \infty, - 1.85122502)$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- 1.85122502, - 0.7109206)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1.28107281$ in the expression.

$f'' (- 1.28107281) = \frac{4 (- 1.28107281) ({(- 1.28107281)}^{8} - 12 {(- 1.28107281)}^{4} + 3)}{{({(- 1.28107281)}^{4} + 1)}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Multiply $4$ by $- 1.28107281$ .

$f'' (- 1.28107281) = \frac{- 5.12429125 ({(- 1.28107281)}^{8} - 12 {(- 1.28107281)}^{4} + 3)}{{({(- 1.28107281)}^{4} + 1)}^{3}}$

Step 6.2.1.2

Multiply $- 5.12429125$ by ${(- 1.28107281)}^{8} - 12 {(- 1.28107281)}^{4} + 3$ .

$f'' (- 1.28107281) = \frac{113.07346647}{{({(- 1.28107281)}^{4} + 1)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Raise $- 1.28107281$ to the power of $4$ .

$f'' (- 1.28107281) = \frac{113.07346647}{{(2.6933653 + 1)}^{3}}$

Step 6.2.2.2

Add $2.6933653$ and $1$ .

$f'' (- 1.28107281) = \frac{113.07346647}{{3.6933653}^{3}}$

Step 6.2.2.3

Raise $3.6933653$ to the power of $3$ .

$f'' (- 1.28107281) = \frac{113.07346647}{50.38100121}$

Step 6.2.3

Divide $113.07346647$ by $50.38100121$ .

$f'' (- 1.28107281) = 2.24436719$

Step 6.2.4

The final answer is $2.24436719$ .

$2.24436719$

Step 6.3

At $- 1.28107281$ , the second derivative is $2.24436719$ . Since this is positive, the second derivative is increasing on the interval $(- 1.85122502, - 0.7109206)$ .

Increasing on $(- 1.85122502, - 0.7109206)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(- 0.7109206, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 0.3554603$ in the expression.

$f'' (- 0.3554603) = \frac{4 (- 0.3554603) ({(- 0.3554603)}^{8} - 12 {(- 0.3554603)}^{4} + 3)}{{({(- 0.3554603)}^{4} + 1)}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Multiply $4$ by $- 0.3554603$ .

$f'' (- 0.3554603) = \frac{- 1.4218412 ({(- 0.3554603)}^{8} - 12 {(- 0.3554603)}^{4} + 3)}{{({(- 0.3554603)}^{4} + 1)}^{3}}$

Step 7.2.1.2

Multiply $- 1.4218412$ by ${(- 0.3554603)}^{8} - 12 {(- 0.3554603)}^{4} + 3$ .

$f'' (- 0.3554603) = \frac{- 3.9934925}{{({(- 0.3554603)}^{4} + 1)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Raise $- 0.3554603$ to the power of $4$ .

$f'' (- 0.3554603) = \frac{- 3.9934925}{{(0.01596483 + 1)}^{3}}$

Step 7.2.2.2

Add $0.01596483$ and $1$ .

$f'' (- 0.3554603) = \frac{- 3.9934925}{{1.01596483}^{3}}$

Step 7.2.2.3

Raise $1.01596483$ to the power of $3$ .

$f'' (- 0.3554603) = \frac{- 3.9934925}{1.0486632}$

Step 7.2.3

Divide $- 3.9934925$ by $1.0486632$ .

$f'' (- 0.3554603) = - 3.80817454$

Step 7.2.4

The final answer is $- 3.80817454$ .

$- 3.80817454$

Step 7.3

At $- 0.3554603$ , the second derivative is $- 3.80817454$ . Since this is negative, the second derivative is decreasing on the interval $(- 0.7109206, 0)$

Decreasing on $(- 0.7109206, 0)$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(0, 0.7109206)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $0.3554603$ in the expression.

$f'' (0.3554603) = \frac{4 (0.3554603) ({(0.3554603)}^{8} - 12 {(0.3554603)}^{4} + 3)}{{({(0.3554603)}^{4} + 1)}^{3}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

Multiply $4$ by $0.3554603$ .

$f'' (0.3554603) = \frac{1.4218412 ({0.3554603}^{8} - 12 \cdot {0.3554603}^{4} + 3)}{{({0.3554603}^{4} + 1)}^{3}}$

Step 8.2.1.2

Multiply $1.4218412$ by ${0.3554603}^{8} - 12 \cdot {0.3554603}^{4} + 3$ .

$f'' (0.3554603) = \frac{3.9934925}{{({0.3554603}^{4} + 1)}^{3}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Raise $0.3554603$ to the power of $4$ .

$f'' (0.3554603) = \frac{3.9934925}{{(0.01596483 + 1)}^{3}}$

Step 8.2.2.2

Add $0.01596483$ and $1$ .

$f'' (0.3554603) = \frac{3.9934925}{{1.01596483}^{3}}$

Step 8.2.2.3

Raise $1.01596483$ to the power of $3$ .

$f'' (0.3554603) = \frac{3.9934925}{1.0486632}$

Step 8.2.3

Divide $3.9934925$ by $1.0486632$ .

$f'' (0.3554603) = 3.80817454$

Step 8.2.4

The final answer is $3.80817454$ .

$3.80817454$

Step 8.3

At $0.3554603$ , the second derivative is $3.80817454$ . Since this is positive, the second derivative is increasing on the interval $(0, 0.7109206)$ .

Increasing on $(0, 0.7109206)$ since $f'' (x) > 0$

Step 9

Substitute a value from the interval $(0.7109206, 1.85122502)$ into the second derivative to determine if it is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $1.28107281$ in the expression.

$f'' (1.28107281) = \frac{4 (1.28107281) ({(1.28107281)}^{8} - 12 {(1.28107281)}^{4} + 3)}{{({(1.28107281)}^{4} + 1)}^{3}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify the numerator.

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Step 9.2.1.1

Multiply $4$ by $1.28107281$ .

$f'' (1.28107281) = \frac{5.12429125 ({1.28107281}^{8} - 12 \cdot {1.28107281}^{4} + 3)}{{({1.28107281}^{4} + 1)}^{3}}$

Step 9.2.1.2

Multiply $5.12429125$ by ${1.28107281}^{8} - 12 \cdot {1.28107281}^{4} + 3$ .

$f'' (1.28107281) = \frac{- 113.07346647}{{({1.28107281}^{4} + 1)}^{3}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Raise $1.28107281$ to the power of $4$ .

$f'' (1.28107281) = \frac{- 113.07346647}{{(2.6933653 + 1)}^{3}}$

Step 9.2.2.2

Add $2.6933653$ and $1$ .

$f'' (1.28107281) = \frac{- 113.07346647}{{3.6933653}^{3}}$

Step 9.2.2.3

Raise $3.6933653$ to the power of $3$ .

$f'' (1.28107281) = \frac{- 113.07346647}{50.38100121}$

Step 9.2.3

Divide $- 113.07346647$ by $50.38100121$ .

$f'' (1.28107281) = - 2.24436719$

Step 9.2.4

The final answer is $- 2.24436719$ .

$- 2.24436719$

Step 9.3

At $1.28107281$ , the second derivative is $- 2.24436719$ . Since this is negative, the second derivative is decreasing on the interval $(0.7109206, 1.85122502)$

Decreasing on $(0.7109206, 1.85122502)$ since $f'' (x) < 0$

Step 10

Substitute a value from the interval $(1.85122502, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 10.1

Replace the variable $x$ with $1.95122502$ in the expression.

$f'' (1.95122502) = \frac{4 (1.95122502) ({(1.95122502)}^{8} - 12 {(1.95122502)}^{4} + 3)}{{({(1.95122502)}^{4} + 1)}^{3}}$

Step 10.2

Simplify the result.

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Step 10.2.1

Simplify the numerator.

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Step 10.2.1.1

Multiply $4$ by $1.95122502$ .

$f'' (1.95122502) = \frac{7.80490009 ({1.95122502}^{8} - 12 \cdot {1.95122502}^{4} + 3)}{{({1.95122502}^{4} + 1)}^{3}}$

Step 10.2.1.2

Multiply $7.80490009$ by ${1.95122502}^{8} - 12 \cdot {1.95122502}^{4} + 3$ .

$f'' (1.95122502) = \frac{305.72871822}{{({1.95122502}^{4} + 1)}^{3}}$

Step 10.2.2

Simplify the denominator.

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Step 10.2.2.1

Raise $1.95122502$ to the power of $4$ .

$f'' (1.95122502) = \frac{305.72871822}{{(14.49537411 + 1)}^{3}}$

Step 10.2.2.2

Add $14.49537411$ and $1$ .

$f'' (1.95122502) = \frac{305.72871822}{{15.49537411}^{3}}$

Step 10.2.2.3

Raise $15.49537411$ to the power of $3$ .

$f'' (1.95122502) = \frac{305.72871822}{3720.54188869}$

Step 10.2.3

Divide $305.72871822$ by $3720.54188869$ .

$f'' (1.95122502) = 0.08217316$

Step 10.2.4

The final answer is $0.08217316$ .

$0.08217316$

Step 10.3

At $1.95122502$ , the second derivative is $0.08217316$ . Since this is positive, the second derivative is increasing on the interval $(1.85122502, \infty)$ .

Increasing on $(1.85122502, \infty)$ since $f'' (x) > 0$

Step 11

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(- 1.85122502, - 0.99559497), (- 0.7109206, - 0.57239819), (0, 0), (0.7109206, 0.57239819), (1.85122502, 0.99559497)$ .

$(- 1.85122502, - 0.99559497), (- 0.7109206, - 0.57239819), (0, 0), (0.7109206, 0.57239819), (1.85122502, 0.99559497)$

Step 12