Calculus Examples

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

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{2 x - 1}{x - 1}$ .

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

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [\frac{2 x - 1}{x - 1}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{2 x - 1}{x - 1}$ .

$\frac{1}{\frac{2 x - 1}{x - 1}} \frac{d}{d x} [\frac{2 x - 1}{x - 1}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{2 x - 1}{x - 1}$ .

$1 \frac{x - 1}{2 x - 1} \frac{d}{d x} [\frac{2 x - 1}{x - 1}]$

Step 3

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

$\frac{x - 1}{2 x - 1} \frac{d}{d x} [\frac{2 x - 1}{x - 1}]$

Step 4

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

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

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

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

Step 5.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 5.3

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

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

Step 5.4

Multiply $2$ by $1$ .

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

Step 5.5

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

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

Step 5.6

Simplify the expression.

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Step 5.6.1

Add $2$ and $0$ .

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

Step 5.6.2

Move $2$ to the left of $x - 1$ .

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

Step 5.7

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

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

Step 5.8

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

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

Step 5.9

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

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

Step 5.10

Combine fractions.

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Step 5.10.1

Add $1$ and $0$ .

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

Step 5.10.2

Multiply $- 1$ by $1$ .

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

Step 5.10.3

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

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

Step 6

Cancel the common factors.

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Step 6.1

Factor $x - 1$ out of $(2 x - 1) {(x - 1)}^{2}$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Simplify the numerator.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Multiply $2$ by $- 1$ .

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

Step 7.3.1.2

Multiply $2$ by $- 1$ .

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

Step 7.3.1.3

Multiply $- 1$ by $- 1$ .

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

Step 7.3.2

Combine the opposite terms in $2 x - 2 - 2 x + 1$ .

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Step 7.3.2.1

Subtract $2 x$ from $2 x$ .

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

Step 7.3.2.2

Subtract $2$ from $0$ .

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

Step 7.3.3

Add $- 2$ and $1$ .

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

Step 7.4

Move the negative in front of the fraction.

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