Calculus Examples

$\frac{\ln (x)}{5 x}$

Step 1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{\ln (x)}{5 x}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [\frac{\ln (x)}{x}]$ .

$\frac{1}{5} \frac{d}{d x} [\frac{\ln (x)}{x}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \ln (x)$ and $g (x) = x$ .

$\frac{1}{5} \cdot \frac{x \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x]}{x^{2}}$

Step 3

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

$\frac{1}{5} \cdot \frac{x \frac{1}{x} - \ln (x) \frac{d}{d x} [x]}{x^{2}}$

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$\frac{1}{5} \cdot \frac{\frac{x}{x} - \ln (x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.2

Cancel the common factor of $x$ .

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Step 4.2.1

Cancel the common factor.

$\frac{1}{5} \cdot \frac{\frac{x}{x} - \ln (x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.2.2

Rewrite the expression.

$\frac{1}{5} \cdot \frac{1 - \ln (x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.3

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

$\frac{1}{5} \cdot \frac{1 - \ln (x) \cdot 1}{x^{2}}$

Step 4.4

Combine fractions.

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Step 4.4.1

Multiply $- 1$ by $1$ .

$\frac{1}{5} \cdot \frac{1 - \ln (x)}{x^{2}}$

Step 4.4.2

Multiply $\frac{1}{5}$ by $\frac{1 - \ln (x)}{x^{2}}$ .

$\frac{1 - \ln (x)}{5 x^{2}}$