Calculus Examples

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

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Differentiate using the Constant Rule.

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Step 5.1

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

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

Step 5.2

Combine fractions.

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Step 5.2.1

Add $\frac{1}{x}$ and $0$ .

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

Step 5.2.2

Combine $\frac{1}{x}$ and $\ln (x)$ .

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

Step 6

To write $(\ln (x) + 1) \frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 7

Combine $(\ln (x) + 1) \frac{1}{x}$ and $\frac{x}{x}$ .

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

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

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

Step 10

Cancel the common factor of $x$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

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

Step 11

Multiply $\ln (x) + 1$ by $1$ .

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

Step 12

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 13

Cancel the common factor of $x$ .

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Step 13.1

Cancel the common factor.

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

Step 13.2

Rewrite the expression.

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

Step 14

Rewrite $\frac{\frac{\ln (1) + 1}{x}}{{(\ln (x) + 1)}^{2}}$ as a product.

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

Step 15

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

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

Step 16

Simplify the numerator.

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Step 16.1

The natural logarithm of $1$ is $0$ .

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

Step 16.2

Add $0$ and $1$ .

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