Calculus Examples

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

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 $f (x) = \ln (x - 1)$ and $g (x) = x - 2$ .

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

Step 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 $g (x) = x - 1$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $x - 2$ by $1$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 3.8

Simplify the expression.

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Step 3.8.1

Add $1$ and $0$ .

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

Step 3.8.2

Multiply $- 1$ by $1$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $x - 2$ by $\frac{1}{x - 1}$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Apply the distributive property.

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

Step 4.1.5.2

Multiply $- \ln (x - 1) \cdot - 1$ .

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Step 4.1.5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.5.2.2

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

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

Step 4.1.6

Reorder factors in $\frac{x - 2 - \ln (x - 1) x + \ln (x - 1)}{x - 1}$ .

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

Step 4.2

Combine terms.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Reorder terms.

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