Calculus Examples

$f (x) = \frac{3}{x}$ at $(9, \frac{1}{3})$

Step 1

Find the first derivative and evaluate at $x = 9$ and $y = \frac{1}{3}$ to find the slope of the tangent line.

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Step 1.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 1.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 1.3

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

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

Step 1.4

Multiply $- 1$ by $3$ .

$- 3 x^{- 2}$

Step 1.5

Simplify.

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Step 1.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.5.2

Combine terms.

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Step 1.5.2.1

Combine $- 3$ and $\frac{1}{x^{2}}$ .

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

Step 1.5.2.2

Move the negative in front of the fraction.

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

Step 1.6

Evaluate the derivative at $x = 9$ .

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

Step 1.7

Simplify.

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Step 1.7.1

Raise $9$ to the power of $2$ .

$- \frac{3}{81}$

Step 1.7.2

Cancel the common factor of $3$ and $81$ .

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Step 1.7.2.1

Factor $3$ out of $3$ .

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

Step 1.7.2.2

Cancel the common factors.

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Step 1.7.2.2.1

Factor $3$ out of $81$ .

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

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

Factor $9$ out of $- 9$ .

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

Step 2.3.1.5.4

Cancel the common factor.

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

Step 2.3.1.5.5

Rewrite the expression.

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

Step 2.3.1.6

Combine $\frac{- 1}{3}$ and $- 1$ .

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

Step 2.3.1.7

Multiply $- 1$ by $- 1$ .

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

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

Add $\frac{1}{3}$ to 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

Add $1$ and $1$ .

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

Step 2.3.2.4

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