Calculus Examples

$f (x) = \frac{1}{x^{3}}$ at $(3, \frac{1}{27})$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = \frac{1}{27}$ to find the slope of the tangent line.

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Step 1.1

Apply basic rules of exponents.

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Step 1.1.1

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

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

Step 1.1.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 1.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.1.2.2

Multiply $3$ by $- 1$ .

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$- 3 x^{- 4}$

Step 1.3

Simplify.

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Step 1.3.1

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

$- 3 \frac{1}{x^{4}}$

Step 1.3.2

Combine terms.

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Step 1.3.2.1

Combine $- 3$ and $\frac{1}{x^{4}}$ .

$\frac{- 3}{x^{4}}$

Step 1.3.2.2

Move the negative in front of the fraction.

$- \frac{3}{x^{4}}$

Step 1.4

Evaluate the derivative at $x = 3$ .

$- \frac{3}{{(3)}^{4}}$

Step 1.5

Simplify.

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Step 1.5.1

Cancel the common factor of $3$ and ${(3)}^{4}$ .

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Step 1.5.1.1

Factor $3$ out of $3$ .

$- \frac{3 (1)}{{(3)}^{4}}$

Step 1.5.1.2

Cancel the common factors.

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Step 1.5.1.2.1

Factor $3$ out of ${(3)}^{4}$ .

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

Step 1.5.1.2.2

Cancel the common factor.

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

Step 1.5.1.2.3

Rewrite the expression.

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

Step 1.5.2

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

$- \frac{1}{27}$

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 $(3, \frac{1}{27})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{1}{27} \cdot (x - 3)$ .

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Combine $x$ and $\frac{1}{27}$ .

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

Step 2.3.1.5

Cancel the common factor of $3$ .

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Step 2.3.1.5.1

Move the leading negative in $- \frac{1}{27}$ into the numerator.

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

Step 2.3.1.5.2

Factor $3$ out of $27$ .

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

Step 2.3.1.5.3

Factor $3$ out of $- 3$ .

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

Step 2.3.1.5.4

Cancel the common factor.

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

Step 2.3.1.5.5

Rewrite the expression.

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

Step 2.3.1.6

Combine $\frac{- 1}{9}$ and $- 1$ .

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

Step 2.3.1.7

Multiply $- 1$ by $- 1$ .

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

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 $\frac{1}{27}$ to both sides of the equation.

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

Step 2.3.2.2

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.3.2.3

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.2.3.1

Multiply $\frac{1}{9}$ by $\frac{3}{3}$ .

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

Step 2.3.2.3.2

Multiply $9$ by $3$ .

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

Step 2.3.2.4

Combine the numerators over the common denominator.

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

Step 2.3.2.5

Add $3$ and $1$ .

$y = - \frac{x}{27} + \frac{4}{27}$

Step 2.3.3

Write in $y = m x + b$ form.

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{1}{27} x) + \frac{4}{27}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{1}{27} x + \frac{4}{27}$

Step 3