Calculus Examples

$\ln (\sqrt{x^{2} + y^{2}})$

Step 1

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as ${(x^{2} + y^{2})}^{\frac{1}{2}}$ .

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

Step 2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 2.3

Replace all occurrences of $u_{1}$ with ${(x^{2} + y^{2})}^{\frac{1}{2}}$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u_{2}$ with $x^{2} + y^{2}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

Combine the numerators over the common denominator.

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

Step 12.4

Add $1$ and $1$ .

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

Step 12.5

Divide $2$ by $2$ .

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

Step 13

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

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

Step 14

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

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

Step 15

Since $x^{2}$ is constant with respect to $y$ , the derivative of $x^{2}$ with respect to $y$ is $0$ .

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

Step 16

Add $0$ and $\frac{d}{d y} [y^{2}]$ .

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

Step 17

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

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

Step 18

Simplify terms.

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Step 18.1

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

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

Step 18.2

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

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

Step 18.3

Cancel the common factor of $2$ .

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Step 18.3.1

Cancel the common factor.

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

Step 18.3.2

Rewrite the expression.

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