Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

The derivative of $\log (u)$ with respect to $u$ is $\frac{1}{u \ln (10)}$ .

$\frac{1}{u \ln (10)} \frac{d}{d x} [x^{2} + y^{2}]$

Step 1.3

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

$\frac{1}{(x^{2} + y^{2}) \ln (10)} \frac{d}{d x} [x^{2} + y^{2}]$

Step 2

Differentiate.

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Step 2.1

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

$\frac{1}{(x^{2} + y^{2}) \ln (10)} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}])$

Step 2.2

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

$\frac{1}{(x^{2} + y^{2}) \ln (10)} (2 x + \frac{d}{d x} [y^{2}])$

Step 2.3

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

$\frac{1}{(x^{2} + y^{2}) \ln (10)} (2 x + 0)$

Step 2.4

Simplify terms.

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Step 2.4.1

Add $2 x$ and $0$ .

$\frac{1}{(x^{2} + y^{2}) \ln (10)} (2 x)$

Step 2.4.2

Combine $2$ and $\frac{1}{(x^{2} + y^{2}) \ln (10)}$ .

$\frac{2}{(x^{2} + y^{2}) \ln (10)} x$

Step 2.4.3

Combine $\frac{2}{(x^{2} + y^{2}) \ln (10)}$ and $x$ .

$\frac{2 x}{(x^{2} + y^{2}) \ln (10)}$

Step 2.4.4

Apply the distributive property.

$\frac{2 x}{x^{2} \ln (10) + y^{2} \ln (10)}$