Calculus Examples

$\frac{1 + \ln (x)}{x}$

Step 1

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 $f (x) = 1 + \ln (x)$ and $g (x) = x$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Add $0$ and $\frac{d}{d x} [\ln (x)]$ .

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

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

Step 4.2

Cancel the common factor of $x$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 4.3

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

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

Step 4.4

Multiply $- 1$ by $1$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Simplify the numerator.

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Step 5.2.1

Multiply $- 1$ by $1$ .

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

Step 5.2.2

Subtract $1$ from $1$ .

$\frac{0 - \ln (x)}{x^{2}}$

Step 5.2.3

Subtract $\ln (x)$ from $0$ .

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

Step 5.3

Move the negative in front of the fraction.

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