Calculus Examples

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

$f (x) = \frac{3}{x}$ at

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

$(- 9, - \frac{1}{3})$

Step 1

Find the first derivative and evaluate at

x = - 9

$x = - 9$ and

y = - \frac{1}{3}

$y = - \frac{1}{3}$ to find the slope of the tangent line.

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Step 1.1

Since

3

$3$ is constant with respect to

x

$x$ , the derivative of

\frac{3}{x}

$\frac{3}{x}$ with respect to

x

$x$ is

3 \frac{d}{d x} [\frac{1}{x}]

$3 \frac{d}{d x} [\frac{1}{x}]$ .

3 \frac{d}{d x} [\frac{1}{x}]

$3 \frac{d}{d x} [\frac{1}{x}]$

Step 1.2

Rewrite

\frac{1}{x}

$\frac{1}{x}$ as

x^{- 1}

$x^{- 1}$ .

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

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

Step 1.3

Differentiate using the Power Rule which states that

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

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

n x^{n - 1}

$n x^{n - 1}$ where

n = - 1

$n = - 1$ .

3 (- x^{- 2})

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

Step 1.4

Multiply

- 1

$- 1$ by

3

$3$ .

- 3 x^{- 2}

$- 3 x^{- 2}$

Step 1.5

Simplify.

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Step 1.5.1

Rewrite the expression using the negative exponent rule

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

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

- 3 \frac{1}{x^{2}}

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

Step 1.5.2

Combine terms.

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Step 1.5.2.1

Combine

- 3

$- 3$ and

\frac{1}{x^{2}}

$\frac{1}{x^{2}}$ .

\frac{- 3}{x^{2}}

$\frac{- 3}{x^{2}}$

Step 1.5.2.2

Move the negative in front of the fraction.

- \frac{3}{x^{2}}

$- \frac{3}{x^{2}}$

- \frac{3}{x^{2}}

$- \frac{3}{x^{2}}$

- \frac{3}{x^{2}}

$- \frac{3}{x^{2}}$

Step 1.6

Evaluate the derivative at

x = - 9

$x = - 9$ .

- \frac{3}{{(- 9)}^{2}}

$- \frac{3}{{(- 9)}^{2}}$

Step 1.7

Simplify.

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Step 1.7.1

Raise

- 9

$- 9$ to the power of

2

$2$ .

- \frac{3}{81}

$- \frac{3}{81}$

Step 1.7.2

Cancel the common factor of

3

$3$ and

81

$81$ .

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Step 1.7.2.1

Factor

3

$3$ out of

3

$3$ .

- \frac{3 (1)}{81}

$- \frac{3 (1)}{81}$

Step 1.7.2.2

Cancel the common factors.

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Step 1.7.2.2.1

Factor

3

$3$ out of

81

$81$ .

- \frac{3 \cdot 1}{3 \cdot 27}

$- \frac{3 \cdot 1}{3 \cdot 27}$

Step 1.7.2.2.2

Cancel the common factor.

$- \frac{3 \cdot 1}{3 \cdot 27}$

Step 1.7.2.2.3

Rewrite the expression.

$- \frac{1}{27}$

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

$- \frac{1}{27}$ and a given point

$(- 9, - \frac{1}{3})$ 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

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

$y - (- \frac{1}{3}) = - \frac{1}{27} \cdot (x - (- 9))$

Step 2.2

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

$y + \frac{1}{3} = - \frac{1}{27} \cdot (x + 9)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$- \frac{1}{27} \cdot (x + 9)$ .

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Step 2.3.1.1

Rewrite.

$y + \frac{1}{3} = 0 + 0 - \frac{1}{27} \cdot (x + 9)$

Step 2.3.1.2

Simplify by adding zeros.

$y + \frac{1}{3} = - \frac{1}{27} \cdot (x + 9)$

Step 2.3.1.3

Apply the distributive property.

$y + \frac{1}{3} = - \frac{1}{27} x - \frac{1}{27} \cdot 9$

Step 2.3.1.4

Combine

$x$ and

$\frac{1}{27}$ .

$y + \frac{1}{3} = - \frac{x}{27} - \frac{1}{27} \cdot 9$

Step 2.3.1.5

Cancel the common factor of

$9$ .

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Step 2.3.1.5.1

Move the leading negative in

$- \frac{1}{27}$ into the numerator.

$y + \frac{1}{3} = - \frac{x}{27} + \frac{- 1}{27} \cdot 9$

Step 2.3.1.5.2

Factor

$9$ out of

$27$ .

$y + \frac{1}{3} = - \frac{x}{27} + \frac{- 1}{9 (3)} \cdot 9$

Step 2.3.1.5.3

Cancel the common factor.

$y + \frac{1}{3} = - \frac{x}{27} + \frac{- 1}{9 \cdot 3} \cdot 9$

Step 2.3.1.5.4

Rewrite the expression.

$y + \frac{1}{3} = - \frac{x}{27} + \frac{- 1}{3}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y + \frac{1}{3} = - \frac{x}{27} - \frac{1}{3}$

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

Subtract

$\frac{1}{3}$ from both sides of the equation.

$y = - \frac{x}{27} - \frac{1}{3} - \frac{1}{3}$

Step 2.3.2.2

Combine the numerators over the common denominator.

$y = \frac{- x}{27} + \frac{- 1 - 1}{3}$

Step 2.3.2.3

Subtract

$1$ from

$- 1$ .

$y = \frac{- x}{27} + \frac{- 2}{3}$

Step 2.3.2.4

Simplify each term.

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Step 2.3.2.4.1

Move the negative in front of the fraction.

$y = - \frac{x}{27} + \frac{- 2}{3}$

Step 2.3.2.4.2

Move the negative in front of the fraction.

$y = - \frac{x}{27} - \frac{2}{3}$

Step 2.3.3

Write in

$y = m x + b$ form.

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{1}{27} x) - \frac{2}{3}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{1}{27} x - \frac{2}{3}$

Step 3