Calculus Examples

$y = \frac{x - 1}{x + 1}$

Step 1

Set $y$ as a function of $x$ .

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

Step 2

Find the derivative.

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Step 2.1

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 2.2

Differentiate.

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Step 2.2.1

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 2.2.2

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 2.2.3

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 2.2.4

Simplify the expression.

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Step 2.2.4.1

Add $1$ and $0$ .

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

Step 2.2.4.2

Multiply $x + 1$ by $1$ .

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

Simplify the expression.

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Step 2.2.8.1

Add $1$ and $0$ .

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

Step 2.2.8.2

Multiply $- 1$ by $1$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Simplify the numerator.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Subtract $x$ from $x$ .

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

Step 2.3.2.1.2

Add $0$ and $1$ .

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

Step 2.3.2.2

Multiply $- 1$ by $- 1$ .

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

Step 2.3.2.3

Add $1$ and $1$ .

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

Step 3

Set the derivative equal to $0$ then solve the equation $\frac{2}{{(x + 1)}^{2}} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$2 = 0$

Step 3.2

Since $2 \neq 0$ , there are no solutions.

No solution

Step 4

There are no solution found by setting the derivative equal to $0$ , $\frac{2}{{(x + 1)}^{2}} = 0$ so there are no horizontal tangent lines.

No horizontal tangent lines found

Step 5