Calculus Examples

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

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 1}$ as ${(x^{2} + 1)}^{\frac{1}{2}}$ .

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

Step 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 $f (x) = {(x^{2} + 1)}^{\frac{1}{2}}$ and $g (x) = x$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

Move ${(x^{2} + 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8.4

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Combine fractions.

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Step 12.1

Add $2 x$ and $0$ .

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

Step 12.2

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

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

Step 12.3

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Add $1$ and $1$ .

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

Step 17

Cancel the common factor.

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

Step 18

Rewrite the expression.

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

Step 19

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

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

Step 20

Combine.

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

Step 21

Apply the distributive property.

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

Step 22

Cancel the common factor of ${(x^{2} + 1)}^{\frac{1}{2}}$ .

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Step 22.1

Cancel the common factor.

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

Step 22.2

Rewrite the expression.

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

Step 23

Multiply ${(x^{2} + 1)}^{\frac{1}{2}}$ by ${(x^{2} + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 23.1

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

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

Step 23.2

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

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

Step 23.3

Combine the numerators over the common denominator.

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

Step 23.4

Add $1$ and $1$ .

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

Step 23.5

Divide $2$ by $2$ .

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

Step 24

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

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

Step 25

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

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

Step 26

Simplify the expression.

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Step 26.1

Multiply $- 1$ by $1$ .

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

Step 26.2

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

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

Step 26.3

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

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

Step 27

Simplify.

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Step 27.1

Apply the distributive property.

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

Step 27.2

Simplify the numerator.

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Step 27.2.1

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

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

Step 27.2.2

Subtract $1 \cdot 1$ from $0$ .

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

Step 27.2.3

Multiply $- 1$ by $1$ .

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

Step 27.3

Move the negative in front of the fraction.

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

Step 27.4

Reorder factors in $- \frac{1}{{(x^{2} + 1)}^{\frac{1}{2}} x^{2}}$ .

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