Calculus Examples

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

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 $g (x) = 3 x^{2} + 2$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $3$ .

$\frac{1}{3 x^{2} + 2} (6 x + \frac{d}{d x} [2])$

Step 2.5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{1}{3 x^{2} + 2} (6 x + 0)$

Step 2.6

Combine fractions.

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Step 2.6.1

Add $6 x$ and $0$ .

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

Step 2.6.2

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

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

Step 2.6.3

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

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