Calculus Examples

$f (x) = \frac{27}{x}$ ; $x = 3$

Step 1

Find the corresponding $y$ -value to $x = 3$ .

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Step 1.1

Substitute $3$ in for $x$ .

$y = \frac{27}{3}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = \frac{27}{3}$

Step 1.2.2

Divide $27$ by $3$ .

$y = 9$

Step 2

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

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Step 2.1

Since $27$ is constant with respect to $x$ , the derivative of $\frac{27}{x}$ with respect to $x$ is $27 \frac{d}{d x} [\frac{1}{x}]$ .

$27 \frac{d}{d x} [\frac{1}{x}]$

Step 2.2

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

$27 \frac{d}{d x} [x^{- 1}]$

Step 2.3

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

$27 (- x^{- 2})$

Step 2.4

Multiply $- 1$ by $27$ .

$- 27 x^{- 2}$

Step 2.5

Simplify.

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Step 2.5.1

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

$- 27 \frac{1}{x^{2}}$

Step 2.5.2

Combine terms.

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Step 2.5.2.1

Combine $- 27$ and $\frac{1}{x^{2}}$ .

$\frac{- 27}{x^{2}}$

Step 2.5.2.2

Move the negative in front of the fraction.

$- \frac{27}{x^{2}}$

Step 2.6

Evaluate the derivative at $x = 3$ .

$- \frac{27}{{(3)}^{2}}$

Step 2.7

Simplify.

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Step 2.7.1

Raise $3$ to the power of $2$ .

$- \frac{27}{9}$

Step 2.7.2

Divide $27$ by $9$ .

$- 1 \cdot 3$

Step 2.7.3

Multiply $- 1$ by $3$ .

$- 3$

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 $- 3$ and a given point $(3, 9)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (9) = - 3 \cdot (x - (3))$

Step 3.2

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

$y - 9 = - 3 \cdot (x - 3)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- 3 \cdot (x - 3)$ .

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Step 3.3.1.1

Rewrite.

$y - 9 = 0 + 0 - 3 \cdot (x - 3)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 9 = - 3 \cdot (x - 3)$

Step 3.3.1.3

Apply the distributive property.

$y - 9 = - 3 x - 3 \cdot - 3$

Step 3.3.1.4

Multiply $- 3$ by $- 3$ .

$y - 9 = - 3 x + 9$

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

Add $9$ to both sides of the equation.

$y = - 3 x + 9 + 9$

Step 3.3.2.2

Add $9$ and $9$ .

$y = - 3 x + 18$

Step 4