Calculus Examples

$y = \frac{1}{3 x^{3}}$ , $(- 2, - \frac{1}{24})$

Step 1

Find the first derivative and evaluate at $x = - 2$ and $y = - \frac{1}{24}$ to find the slope of the tangent line.

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Step 1.1

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

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

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.2.2

Multiply $3$ by $- 1$ .

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

Step 1.3

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

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

Step 1.4

Simplify terms.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4.4

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

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Step 1.4.4.1

Factor $3$ out of $- 3$ .

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

Step 1.4.4.2

Cancel the common factors.

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Step 1.4.4.2.1

Factor $3$ out of $3 x^{4}$ .

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

Step 1.4.4.2.2

Cancel the common factor.

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

Step 1.4.4.2.3

Rewrite the expression.

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

Step 1.4.5

Move the negative in front of the fraction.

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

Step 1.5

Evaluate the derivative at $x = - 2$ .

$- \frac{1}{{(- 2)}^{4}}$

Step 1.6

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

$- \frac{1}{16}$

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 $- \frac{1}{16}$ and a given point $(- 2, - \frac{1}{24})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- \frac{1}{24}) = - \frac{1}{16} \cdot (x - (- 2))$

Step 2.2

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

$y + \frac{1}{24} = - \frac{1}{16} \cdot (x + 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{1}{16} \cdot (x + 2)$ .

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Step 2.3.1.1

Rewrite.

$y + \frac{1}{24} = 0 + 0 - \frac{1}{16} \cdot (x + 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y + \frac{1}{24} = - \frac{1}{16} \cdot (x + 2)$

Step 2.3.1.3

Apply the distributive property.

$y + \frac{1}{24} = - \frac{1}{16} x - \frac{1}{16} \cdot 2$

Step 2.3.1.4

Combine $x$ and $\frac{1}{16}$ .

$y + \frac{1}{24} = - \frac{x}{16} - \frac{1}{16} \cdot 2$

Step 2.3.1.5

Cancel the common factor of $2$ .

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Step 2.3.1.5.1

Move the leading negative in $- \frac{1}{16}$ into the numerator.

$y + \frac{1}{24} = - \frac{x}{16} + \frac{- 1}{16} \cdot 2$

Step 2.3.1.5.2

Factor $2$ out of $16$ .

$y + \frac{1}{24} = - \frac{x}{16} + \frac{- 1}{2 (8)} \cdot 2$

Step 2.3.1.5.3

Cancel the common factor.

$y + \frac{1}{24} = - \frac{x}{16} + \frac{- 1}{2 \cdot 8} \cdot 2$

Step 2.3.1.5.4

Rewrite the expression.

$y + \frac{1}{24} = - \frac{x}{16} + \frac{- 1}{8}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y + \frac{1}{24} = - \frac{x}{16} - \frac{1}{8}$

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 $\frac{1}{24}$ from both sides of the equation.

$y = - \frac{x}{16} - \frac{1}{8} - \frac{1}{24}$

Step 2.3.2.2

To write $- \frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{x}{16} - \frac{1}{8} \cdot \frac{3}{3} - \frac{1}{24}$

Step 2.3.2.3

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.2.3.1

Multiply $\frac{1}{8}$ by $\frac{3}{3}$ .

$y = - \frac{x}{16} - \frac{3}{8 \cdot 3} - \frac{1}{24}$

Step 2.3.2.3.2

Multiply $8$ by $3$ .

$y = - \frac{x}{16} - \frac{3}{24} - \frac{1}{24}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{x}{16} + \frac{- 3 - 1}{24}$

Step 2.3.2.5

Subtract $1$ from $- 3$ .

$y = - \frac{x}{16} + \frac{- 4}{24}$

Step 2.3.2.6

Simplify each term.

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Step 2.3.2.6.1

Cancel the common factor of $- 4$ and $24$ .

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Step 2.3.2.6.1.1

Factor $4$ out of $- 4$ .

$y = - \frac{x}{16} + \frac{4 (- 1)}{24}$

Step 2.3.2.6.1.2

Cancel the common factors.

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Step 2.3.2.6.1.2.1

Factor $4$ out of $24$ .

$y = - \frac{x}{16} + \frac{4 \cdot - 1}{4 \cdot 6}$

Step 2.3.2.6.1.2.2

Cancel the common factor.

$y = - \frac{x}{16} + \frac{4 \cdot - 1}{4 \cdot 6}$

Step 2.3.2.6.1.2.3

Rewrite the expression.

$y = - \frac{x}{16} + \frac{- 1}{6}$

Step 2.3.2.6.2

Move the negative in front of the fraction.

$y = - \frac{x}{16} - \frac{1}{6}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{1}{16} x) - \frac{1}{6}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{1}{16} x - \frac{1}{6}$

Step 3