Calculus Examples

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

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Since $2$ is constant with respect to $y$ , the derivative of $2 y^{2}$ with respect to $y$ is $2 \frac{d}{d y} [y^{2}]$ .

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Combine fractions.

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Step 2.9.1

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

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

Step 2.9.2

Multiply $2$ by $2$ .

$\frac{4}{1 + x^{2} + 2 y^{2}} y$

Step 2.9.3

Combine $\frac{4}{1 + x^{2} + 2 y^{2}}$ and $y$ .

$\frac{4 y}{1 + x^{2} + 2 y^{2}}$

Step 2.9.4

Reorder terms.

$\frac{4 y}{x^{2} + 2 y^{2} + 1}$