Calculus Examples

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

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

$\frac{1}{(1 + x^{2}) \ln (10)} (0 + \frac{d}{d x} [x^{2}])$

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine fractions.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $\ln (10)$ by $1$ .

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

Step 3.3

Reorder terms.

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