Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $1$ and $0$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Add $1$ and $0$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $x$ by $1$ .

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

Step 3.4.1.2

Multiply $- 1$ by $1$ .

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

Step 3.4.1.3

Multiply $- 1$ by $1$ .

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

Step 3.4.1.4

Multiply $- 1 \cdot 1 \cdot 1$ .

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Step 3.4.1.4.1

Multiply $- 1$ by $1$ .

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

Step 3.4.1.4.2

Multiply $- 1$ by $1$ .

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

Step 3.4.2

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

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Step 3.4.2.1

Subtract $x$ from $x$ .

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

Step 3.4.2.2

Subtract $1$ from $0$ .

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

Step 3.4.3

Subtract $1$ from $- 1$ .

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

Step 3.5

Move the negative in front of the fraction.

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