Calculus Examples

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

Step 1

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 $g (y) = x^{2} + y^{2}$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine fractions.

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Step 2.5.1

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

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

Step 2.5.2

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

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