Algebra Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $2 x$ and $0$ .

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

Step 1.2.6.2

Multiply $2$ by $- 1$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Add $1$ and $2$ .

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

Step 1.6

Simplify.

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Step 1.6.1

Apply the distributive property.

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

Step 1.6.2

Apply the distributive property.

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

Step 1.6.3

Simplify the numerator.

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Step 1.6.3.1

Simplify each term.

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Step 1.6.3.1.1

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

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Step 1.6.3.1.1.1

Move $x$ .

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

Step 1.6.3.1.1.2

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

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Step 1.6.3.1.1.2.1

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

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

Step 1.6.3.1.1.2.2

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

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

Step 1.6.3.1.1.3

Add $1$ and $2$ .

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

Step 1.6.3.1.2

Multiply $2$ by $1$ .

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

Step 1.6.3.2

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

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Step 1.6.3.2.1

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

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

Step 1.6.3.2.2

Add $2 x$ and $0$ .

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

Multiply $2$ by $2$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

Simplify with factoring out.

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Step 2.5.1

Multiply $2$ by $- 1$ .

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

Step 2.5.2

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

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Step 2.5.2.1

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

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

Step 2.5.2.2

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

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

Step 2.5.2.3

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

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

Step 2.6

Cancel the common factors.

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Step 2.6.1

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

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

Step 2.6.2

Cancel the common factor.

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

Step 2.6.3

Rewrite the expression.

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2 x$ and $0$ .

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

Step 2.10.2

Multiply $2$ by $- 2$ .

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

Add $1$ and $1$ .

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

Simplify.

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Step 2.17.1

Apply the distributive property.

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

Step 2.17.2

Simplify each term.

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Step 2.17.2.1

Multiply $- 3$ by $2$ .

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

Step 2.17.2.2

Multiply $2$ by $1$ .

$f'' (x) = \frac{- 6 x^{2} + 2}{{(x^{2} + 1)}^{3}}$

Step 2.17.3

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

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Step 2.17.3.1

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

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

Step 2.17.3.2

Factor $2$ out of $2$ .

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

Step 2.17.3.3

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

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

Step 2.17.4

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

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

Step 2.17.5

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

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

Step 2.17.6

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

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

Step 2.17.7

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

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

Step 2.17.8

Move the negative in front of the fraction.

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