Calculus Examples

$y = \frac{1}{x^{3}}$ , $(- 2, - \frac{1}{8})$

Step 1

Find the first derivative and evaluate at $x = - 2$ and $y = - \frac{1}{8}$ 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 $x = - 2$ .

$- \frac{3}{{(- 2)}^{4}}$

Step 1.5

Raise $- 2$ to the power of $4$ .

$- \frac{3}{16}$

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 $- \frac{3}{16}$ and a given point $(- 2, - \frac{1}{8})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- \frac{1}{8}) = - \frac{3}{16} \cdot (x - (- 2))$

Step 2.2

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

$y + \frac{1}{8} = - \frac{3}{16} \cdot (x + 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{3}{16} \cdot (x + 2)$ .

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Step 2.3.1.1

Rewrite.

$y + \frac{1}{8} = 0 + 0 - \frac{3}{16} \cdot (x + 2)$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

$y + \frac{1}{8} = - \frac{3}{16} x - \frac{3}{16} \cdot 2$

Step 2.3.1.2.2

Combine $x$ and $\frac{3}{16}$ .

$y + \frac{1}{8} = - \frac{x \cdot 3}{16} - \frac{3}{16} \cdot 2$

Step 2.3.1.2.3

Cancel the common factor of $2$ .

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Step 2.3.1.2.3.1

Move the leading negative in $- \frac{3}{16}$ into the numerator.

$y + \frac{1}{8} = - \frac{x \cdot 3}{16} + \frac{- 3}{16} \cdot 2$

Step 2.3.1.2.3.2

Factor $2$ out of $16$ .

$y + \frac{1}{8} = - \frac{x \cdot 3}{16} + \frac{- 3}{2 (8)} \cdot 2$

Step 2.3.1.2.3.3

Cancel the common factor.

$y + \frac{1}{8} = - \frac{x \cdot 3}{16} + \frac{- 3}{2 \cdot 8} \cdot 2$

Step 2.3.1.2.3.4

Rewrite the expression.

$y + \frac{1}{8} = - \frac{x \cdot 3}{16} + \frac{- 3}{8}$

Step 2.3.1.3

Simplify each term.

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Step 2.3.1.3.1

Move $3$ to the left of $x$ .

$y + \frac{1}{8} = - \frac{3 \cdot x}{16} + \frac{- 3}{8}$

Step 2.3.1.3.2

Move the negative in front of the fraction.

$y + \frac{1}{8} = - \frac{3 x}{16} - \frac{3}{8}$

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 $\frac{1}{8}$ from both sides of the equation.

$y = - \frac{3 x}{16} - \frac{3}{8} - \frac{1}{8}$

Step 2.3.2.2

Combine the numerators over the common denominator.

$y = \frac{- 3 x}{16} + \frac{- 3 - 1}{8}$

Step 2.3.2.3

Subtract $1$ from $- 3$ .

$y = \frac{- 3 x}{16} + \frac{- 4}{8}$

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.

$y = - \frac{3 x}{16} + \frac{- 4}{8}$

Step 2.3.2.4.2

Cancel the common factor of $- 4$ and $8$ .

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Step 2.3.2.4.2.1

Factor $4$ out of $- 4$ .

$y = - \frac{3 x}{16} + \frac{4 (- 1)}{8}$

Step 2.3.2.4.2.2

Cancel the common factors.

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Step 2.3.2.4.2.2.1

Factor $4$ out of $8$ .

$y = - \frac{3 x}{16} + \frac{4 \cdot - 1}{4 \cdot 2}$

Step 2.3.2.4.2.2.2

Cancel the common factor.

$y = - \frac{3 x}{16} + \frac{4 \cdot - 1}{4 \cdot 2}$

Step 2.3.2.4.2.2.3

Rewrite the expression.

$y = - \frac{3 x}{16} + \frac{- 1}{2}$

Step 2.3.2.4.3

Move the negative in front of the fraction.

$y = - \frac{3 x}{16} - \frac{1}{2}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{3}{16} x) - \frac{1}{2}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{3}{16} x - \frac{1}{2}$

Step 3