Calculus Examples

y = \frac{2 x}{x - 1}

$y = \frac{2 x}{x - 1}$ at

(- 1, 1)

$(- 1, 1)$

Step 1

Find the first derivative and evaluate at

x = - 1

$x = - 1$ and

y = 1

$y = 1$ to find the slope of the tangent line.

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Step 1.1

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

\frac{2 x}{x - 1}

$\frac{2 x}{x - 1}$ with respect to

x

$x$ is

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

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

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

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

Step 1.2

Differentiate using the Quotient Rule which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}]

$\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}

$\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where

f (x) = x

$f (x) = x$ and

g (x) = x - 1

$g (x) = x - 1$ .

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

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

Step 1.3

Differentiate.

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Step 1.3.1

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.3.2

Multiply

$x - 1$ by

$1$ .

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

Step 1.3.3

By the Sum Rule, the derivative of

$x - 1$ with respect to

$x$ is

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

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

Step 1.3.4

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.3.5

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- 1$ with respect to

$x$ is

$0$ .

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

Step 1.3.6

Simplify terms.

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Step 1.3.6.1

Add

$1$ and

$0$ .

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

Step 1.3.6.2

Multiply

$- 1$ by

$1$ .

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

Step 1.3.6.3

Subtract

$x$ from

$x$ .

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

Step 1.3.6.4

Simplify the expression.

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Step 1.3.6.4.1

Subtract

$1$ from

$0$ .

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

Step 1.3.6.4.2

Move the negative in front of the fraction.

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

Step 1.3.6.4.3

Multiply

$- 1$ by

$2$ .

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

Step 1.3.6.5

Combine

$- 2$ and

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

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

Step 1.3.6.6

Move the negative in front of the fraction.

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

Step 1.4

Evaluate the derivative at

$x = - 1$ .

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

Step 1.5

Simplify.

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Step 1.5.1

Simplify the denominator.

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Step 1.5.1.1

Subtract

$1$ from

$- 1$ .

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

Step 1.5.1.2

Raise

$- 2$ to the power of

$2$ .

$- \frac{2}{4}$

Step 1.5.2

Cancel the common factor of

$2$ and

$4$ .

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Step 1.5.2.1

Factor

$2$ out of

$2$ .

$- \frac{2 (1)}{4}$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Factor

$2$ out of

$4$ .

$- \frac{2 \cdot 1}{2 \cdot 2}$

Step 1.5.2.2.2

Cancel the common factor.

$- \frac{2 \cdot 1}{2 \cdot 2}$

Step 1.5.2.2.3

Rewrite the expression.

$- \frac{1}{2}$

Step 2

Plug the slope and point values into the point-slope formula and solve for

$y$ .

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Step 2.1

Use the slope

$- \frac{1}{2}$ and a given point

$(- 1, 1)$ to substitute for

$x_{1}$ and

$y_{1}$ in the point-slope form

$y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

Simplify the equation and keep it in point-slope form.

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

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

$y - 1 = - \frac{1}{2} x - \frac{1}{2} \cdot 1$

Step 2.3.1.4

Combine

$x$ and

$\frac{1}{2}$ .

$y - 1 = - \frac{x}{2} - \frac{1}{2} \cdot 1$

Step 2.3.1.5

Multiply

$- 1$ by

$1$ .

$y - 1 = - \frac{x}{2} - \frac{1}{2}$

Step 2.3.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 2.3.2.1

Add

$1$ to both sides of the equation.

$y = - \frac{x}{2} - \frac{1}{2} + 1$

Step 2.3.2.2

Write

$1$ as a fraction with a common denominator.

$y = - \frac{x}{2} - \frac{1}{2} + \frac{2}{2}$

Step 2.3.2.3

Combine the numerators over the common denominator.

$y = - \frac{x}{2} + \frac{- 1 + 2}{2}$

Step 2.3.2.4

Add

$- 1$ and

$2$ .

$y = - \frac{x}{2} + \frac{1}{2}$

Step 2.3.3

Write in

$y = m x + b$ form.

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Step 2.3.3.1

Reorder terms.

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

Step 2.3.3.2

Remove parentheses.

$y = - \frac{1}{2} x + \frac{1}{2}$

Step 3