Calculus Examples

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

Step 1

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

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

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify.

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

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

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

Step 10.2

Subtract $2$ from $1$ .

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

Step 11

Combine fractions.

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Step 11.1

Move the negative in front of the fraction.

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

Step 11.2

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

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

Step 11.3

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

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

Step 11.4

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Combine fractions.

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Step 15.1

Add $2 x$ and $0$ .

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

Step 15.2

Multiply $2$ by $- 1$ .

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

Step 15.3

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

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

Step 15.4

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

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

Step 16

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

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

Step 17

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

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

Step 18

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

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

Step 19

Add $1$ and $1$ .

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

Step 20

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

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

Step 21

Cancel the common factors.

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Step 21.1

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

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

Step 21.2

Cancel the common factor.

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

Step 21.3

Rewrite the expression.

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

Step 22

Move the negative in front of the fraction.

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

Step 23

To write ${(x^{2} + 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x^{2} + 1)}^{\frac{1}{2}}}{{(x^{2} + 1)}^{\frac{1}{2}}}$ .

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

Step 24

Combine the numerators over the common denominator.

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

Step 25

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

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Step 25.1

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

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

Step 25.2

Combine the numerators over the common denominator.

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

Step 25.3

Add $1$ and $1$ .

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

Step 25.4

Divide $2$ by $2$ .

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

Step 26

Simplify ${(x^{2} + 1)}^{1}$ .

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

Step 27

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

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

Step 28

Add $0$ and $1$ .

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

Step 29

Rewrite $\frac{\frac{1}{{(x^{2} + 1)}^{\frac{1}{2}}}}{x^{2} + 1}$ as a product.

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

Step 30

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

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

Step 31

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

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Step 31.1

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

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Step 31.1.1

Raise $x^{2} + 1$ to the power of $1$ .

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

Step 31.1.2

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

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

Step 31.2

Write $1$ as a fraction with a common denominator.

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

Step 31.3

Combine the numerators over the common denominator.

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

Step 31.4

Add $1$ and $2$ .

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