Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \frac{x + 1}{x - 1}$ .

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Step 1.1

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

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 1.3

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

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

Step 2

Combine $2$ and $\frac{x + 1}{x - 1}$ .

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

Step 3

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 $g (x) = x - 1$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $1$ and $0$ .

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

Step 4.4.2

Multiply $x - 1$ by $1$ .

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

Combine fractions.

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Step 4.8.1

Add $1$ and $0$ .

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

Step 4.8.2

Multiply $- 1$ by $1$ .

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

Step 4.8.3

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

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

Step 5

Multiply $x - 1$ by ${(x - 1)}^{2}$ by adding the exponents.

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Step 5.1

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

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Step 5.1.1

Raise $x - 1$ to the power of $1$ .

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

Step 5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.2

Add $1$ and $2$ .

$\frac{2 (x + 1) (x - 1 - (x + 1))}{{(x - 1)}^{3}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{(2 x + 2 \cdot 1) (x - 1 - (x + 1))}{{(x - 1)}^{3}}$

Step 6.2

Apply the distributive property.

$\frac{(2 x + 2 \cdot 1) (x - 1 - x - 1 \cdot 1)}{{(x - 1)}^{3}}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Multiply $2$ by $1$ .

$\frac{(2 x + 2) (x - 1 - x - 1 \cdot 1)}{{(x - 1)}^{3}}$

Step 6.3.2

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

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Step 6.3.2.1

Subtract $x$ from $x$ .

$\frac{(2 x + 2) (0 - 1 - 1 \cdot 1)}{{(x - 1)}^{3}}$

Step 6.3.2.2

Subtract $1$ from $0$ .

$\frac{(2 x + 2) (- 1 - 1 \cdot 1)}{{(x - 1)}^{3}}$

Step 6.3.3

Multiply $- 1$ by $1$ .

$\frac{(2 x + 2) (- 1 - 1)}{{(x - 1)}^{3}}$

Step 6.3.4

Subtract $1$ from $- 1$ .

$\frac{(2 x + 2) \cdot - 2}{{(x - 1)}^{3}}$

Step 6.3.5

Apply the distributive property.

$\frac{2 x \cdot - 2 + 2 \cdot - 2}{{(x - 1)}^{3}}$

Step 6.3.6

Multiply $- 2$ by $2$ .

$\frac{- 4 x + 2 \cdot - 2}{{(x - 1)}^{3}}$

Step 6.3.7

Multiply $2$ by $- 2$ .

$\frac{- 4 x - 4}{{(x - 1)}^{3}}$

Step 6.4

Factor $4$ out of $- 4 x - 4$ .

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Step 6.4.1

Factor $4$ out of $- 4 x$ .

$\frac{4 (- x) - 4}{{(x - 1)}^{3}}$

Step 6.4.2

Factor $4$ out of $- 4$ .

$\frac{4 (- x) + 4 (- 1)}{{(x - 1)}^{3}}$

Step 6.4.3

Factor $4$ out of $4 (- x) + 4 (- 1)$ .

$\frac{4 (- x - 1)}{{(x - 1)}^{3}}$

Step 6.5

Factor $- 1$ out of $- x$ .

$\frac{4 (- (x) - 1)}{{(x - 1)}^{3}}$

Step 6.6

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{4 (- (x) - 1 (1))}{{(x - 1)}^{3}}$

Step 6.7

Factor $- 1$ out of $- (x) - 1 (1)$ .

$\frac{4 (- (x + 1))}{{(x - 1)}^{3}}$

Step 6.8

Rewrite $- (x + 1)$ as $- 1 (x + 1)$ .

$\frac{4 (- 1 (x + 1))}{{(x - 1)}^{3}}$

Step 6.9

Move the negative in front of the fraction.

$- \frac{4 (x + 1)}{{(x - 1)}^{3}}$