Calculus Examples

$\ln (1 + 3 x^{2})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 1 + 3 x^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $1 + 3 x^{2}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [1 + 3 x^{2}]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [1 + 3 x^{2}]$

Step 1.3

Replace all occurrences of $u$ with $1 + 3 x^{2}$ .

$\frac{1}{1 + 3 x^{2}} \frac{d}{d x} [1 + 3 x^{2}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $1 + 3 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [3 x^{2}]$ .

$\frac{1}{1 + 3 x^{2}} (\frac{d}{d x} [1] + \frac{d}{d x} [3 x^{2}])$

Step 2.2

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

$\frac{1}{1 + 3 x^{2}} (0 + \frac{d}{d x} [3 x^{2}])$

Step 2.3

Add $0$ and $\frac{d}{d x} [3 x^{2}]$ .

$\frac{1}{1 + 3 x^{2}} \frac{d}{d x} [3 x^{2}]$

Step 2.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{1 + 3 x^{2}} (3 \frac{d}{d x} [x^{2}])$

Step 2.5

Combine $3$ and $\frac{1}{1 + 3 x^{2}}$ .

$\frac{3}{1 + 3 x^{2}} \frac{d}{d x} [x^{2}]$

Step 2.6

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

$\frac{3}{1 + 3 x^{2}} (2 x)$

Step 2.7

Combine fractions.

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Step 2.7.1

Combine $2$ and $\frac{3}{1 + 3 x^{2}}$ .

$\frac{2 \cdot 3}{1 + 3 x^{2}} x$

Step 2.7.2

Multiply $2$ by $3$ .

$\frac{6}{1 + 3 x^{2}} x$

Step 2.7.3

Combine $\frac{6}{1 + 3 x^{2}}$ and $x$ .

$\frac{6 x}{1 + 3 x^{2}}$

Step 2.7.4

Reorder terms.

$\frac{6 x}{3 x^{2} + 1}$