Calculus Examples

f (x) = \frac{1}{x - 3}

$f (x) = \frac{1}{x - 3}$ ;

x = - 1

$x = - 1$

Step 1

Find the corresponding

y

$y$ -value to

x = - 1

$x = - 1$ .

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Step 1.1

Substitute

- 1

$- 1$ in for

x

$x$ .

y = \frac{1}{(- 1) - 3}

$y = \frac{1}{(- 1) - 3}$

Step 1.2

Solve for

y

$y$ .

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Step 1.2.1

Remove parentheses.

y = \frac{1}{- 1 - 3}

$y = \frac{1}{- 1 - 3}$

Step 1.2.2

Remove parentheses.

y = \frac{1}{(- 1) - 3}

$y = \frac{1}{(- 1) - 3}$

Step 1.2.3

Simplify

\frac{1}{(- 1) - 3}

$\frac{1}{(- 1) - 3}$ .

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Step 1.2.3.1

Subtract

3

$3$ from

- 1

$- 1$ .

y = \frac{1}{- 4}

$y = \frac{1}{- 4}$

Step 1.2.3.2

Move the negative in front of the fraction.

y = - \frac{1}{4}

$y = - \frac{1}{4}$

y = - \frac{1}{4}

$y = - \frac{1}{4}$

y = - \frac{1}{4}

$y = - \frac{1}{4}$

y = - \frac{1}{4}

$y = - \frac{1}{4}$

Step 2

Find the first derivative and evaluate at

x = - 1

$x = - 1$ and

y = - \frac{1}{4}

$y = - \frac{1}{4}$ to find the slope of the tangent line.

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Step 2.1

Rewrite

\frac{1}{x - 3}

$\frac{1}{x - 3}$ as

{(x - 3)}^{- 1}

${(x - 3)}^{- 1}$ .

\frac{d}{d x} [{(x - 3)}^{- 1}]

$\frac{d}{d x} [{(x - 3)}^{- 1}]$

Step 2.2

Differentiate using the chain rule, which states that

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

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

$f' (g (x)) g' (x)$ where

$f (x) = x^{- 1}$ and

$g (x) = x - 3$ .

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Step 2.2.1

To apply the Chain Rule, set

$u$ as

$x - 3$ .

$\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x - 3]$

Step 2.2.2

Differentiate using the Power Rule which states that

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

$n u^{n - 1}$ where

$n = - 1$ .

$- u^{- 2} \frac{d}{d x} [x - 3]$

Step 2.2.3

Replace all occurrences of

$u$ with

$x - 3$ .

$- {(x - 3)}^{- 2} \frac{d}{d x} [x - 3]$

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of

$x - 3$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$- {(x - 3)}^{- 2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 2.3.2

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

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

Step 2.3.3

Since

$- 3$ is constant with respect to

$x$ , the derivative of

$- 3$ with respect to

$x$ is

$0$ .

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

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

Add

$1$ and

$0$ .

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

Step 2.3.4.2

Multiply

$- 1$ by

$1$ .

$- {(x - 3)}^{- 2}$

Step 2.4

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.5

Evaluate the derivative at

$x = - 1$ .

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

Step 2.6

Simplify the denominator.

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Step 2.6.1

Subtract

$3$ from

$- 1$ .

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

Step 2.6.2

Raise

$- 4$ to the power of

$2$ .

$- \frac{1}{16}$

Step 3

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

$y$ .

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Step 3.1

Use the slope

$- \frac{1}{16}$ and a given point

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

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

$y - (- \frac{1}{4}) = - \frac{1}{16} \cdot (x - (- 1))$

Step 3.2

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

$y + \frac{1}{4} = - \frac{1}{16} \cdot (x + 1)$

Step 3.3

Solve for

$y$ .

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Step 3.3.1

Simplify

$- \frac{1}{16} \cdot (x + 1)$ .

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Step 3.3.1.1

Rewrite.

$y + \frac{1}{4} = 0 + 0 - \frac{1}{16} \cdot (x + 1)$

Step 3.3.1.2

Simplify by adding zeros.

$y + \frac{1}{4} = - \frac{1}{16} \cdot (x + 1)$

Step 3.3.1.3

Apply the distributive property.

$y + \frac{1}{4} = - \frac{1}{16} x - \frac{1}{16} \cdot 1$

Step 3.3.1.4

Combine

$x$ and

$\frac{1}{16}$ .

$y + \frac{1}{4} = - \frac{x}{16} - \frac{1}{16} \cdot 1$

Step 3.3.1.5

Multiply

$- 1$ by

$1$ .

$y + \frac{1}{4} = - \frac{x}{16} - \frac{1}{16}$

Step 3.3.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 3.3.2.1

Subtract

$\frac{1}{4}$ from both sides of the equation.

$y = - \frac{x}{16} - \frac{1}{16} - \frac{1}{4}$

Step 3.3.2.2

To write

$- \frac{1}{4}$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$y = - \frac{x}{16} - \frac{1}{16} - \frac{1}{4} \cdot \frac{4}{4}$

Step 3.3.2.3

Write each expression with a common denominator of

$16$ , by multiplying each by an appropriate factor of

$1$ .

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Step 3.3.2.3.1

Multiply

$\frac{1}{4}$ by

$\frac{4}{4}$ .

$y = - \frac{x}{16} - \frac{1}{16} - \frac{4}{4 \cdot 4}$

Step 3.3.2.3.2

Multiply

$4$ by

$4$ .

$y = - \frac{x}{16} - \frac{1}{16} - \frac{4}{16}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{x}{16} + \frac{- 1 - 4}{16}$

Step 3.3.2.5

Subtract

$4$ from

$- 1$ .

$y = - \frac{x}{16} + \frac{- 5}{16}$

Step 3.3.2.6

Move the negative in front of the fraction.

$y = - \frac{x}{16} - \frac{5}{16}$

Step 3.3.3

Write in

$y = m x + b$ form.

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Step 3.3.3.1

Reorder terms.

$y = - (\frac{1}{16} x) - \frac{5}{16}$

Step 3.3.3.2

Remove parentheses.

$y = - \frac{1}{16} x - \frac{5}{16}$

Step 4