Calculus Examples

$y = \frac{1 + \ln (x)}{1 - \ln (x)}$

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 $g (x) = 1 - \ln (x)$ .

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

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Multiply.

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Step 4.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.5.2

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Simplify the numerator.

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Step 6.3.1

Combine the opposite terms in $1 \frac{1}{x} - \ln (x) \frac{1}{x} + 1 \frac{1}{x} + \ln (x) \frac{1}{x}$ .

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Step 6.3.1.1

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

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

Step 6.3.1.2

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

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

Step 6.3.2

Simplify each term.

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Step 6.3.2.1

Multiply $\frac{1}{x}$ by $1$ .

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

Step 6.3.2.2

Multiply $\frac{1}{x}$ by $1$ .

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

Step 6.3.3

Combine the numerators over the common denominator.

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

Step 6.3.4

Add $1$ and $1$ .

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

Step 6.4

Combine terms.

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Step 6.4.1

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

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

Step 6.4.2

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

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