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

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{3}}$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

Multiply $3$ by $- 1$ .

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

Step 1.2.3

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

$- (- 3 x^{- 4}) + \frac{1}{x^{3}} \frac{d}{d x} [- 1]$

Step 1.2.4

Multiply $- 3$ by $- 1$ .

$3 x^{- 4} + \frac{1}{x^{3}} \frac{d}{d x} [- 1]$

Step 1.2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$3 x^{- 4} + \frac{1}{x^{3}} \cdot 0$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Multiply $\frac{1}{x^{3}}$ by $0$ .

$3 x^{- 4} + 0$

Step 1.2.6.2

Add $3 x^{- 4}$ and $0$ .

$3 x^{- 4}$

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

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

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 by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

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

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

Step 2.3.1.5

Cancel the common factor of $2$ .

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Step 2.3.1.5.1

Factor $2$ out of $16$ .

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

Step 2.3.1.5.2

Cancel the common factor.

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

Step 2.3.1.5.3

Rewrite the expression.

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

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

Add $\frac{1}{8}$ to 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

Add $3$ and $1$ .

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

Step 2.3.2.4

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

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Step 2.3.2.4.1

Factor $4$ out of $4$ .

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

Step 2.3.2.4.2

Cancel the common factors.

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Step 2.3.2.4.2.1

Factor $4$ out of $8$ .

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

Step 2.3.2.4.2.2

Cancel the common factor.

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

Step 2.3.2.4.2.3

Rewrite the expression.

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

Step 2.3.3

Reorder terms.

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

Step 3