Calculus Examples

$x^{2} - \ln (3 x)$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Evaluate $\frac{d}{d x} [- \ln (3 x)]$ .

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (3 x)$ with respect to $x$ is $- \frac{d}{d x} [\ln (3 x)]$ .

$2 x - \frac{d}{d x} [\ln (3 x)]$

Step 2.2

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 $g (x) = 3 x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $3 x$ .

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

Step 2.2.2

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

$2 x - (\frac{1}{u} \frac{d}{d x} [3 x])$

Step 2.2.3

Replace all occurrences of $u$ with $3 x$ .

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

Step 2.3

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

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

Step 2.4

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

$2 x - (\frac{1}{3 x} (3 \cdot 1))$

Step 2.5

Multiply $3$ by $1$ .

$2 x - (\frac{1}{3 x} \cdot 3)$

Step 2.6

Combine $\frac{1}{3 x}$ and $3$ .

$2 x - \frac{3}{3 x}$

Step 2.7

Cancel the common factor of $3$ .

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Step 2.7.1

Cancel the common factor.

$2 x - \frac{3}{3 x}$

Step 2.7.2

Rewrite the expression.

$2 x - \frac{1}{x}$