Calculus Examples

y = \frac{1}{3 x^{3}}

$y = \frac{1}{3 x^{3}}$ ,

$(2, \frac{1}{24})$

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

Factor

$2$ out of

$- 2$ .

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

Step 2.3.1.5.4

Cancel the common factor.

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

Step 2.3.1.5.5

Rewrite the expression.

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

Step 2.3.1.6

Combine

$\frac{- 1}{8}$ and

$- 1$ .

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

Step 2.3.1.7

Multiply

$- 1$ by

$- 1$ .

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

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

Add

$\frac{1}{24}$ to 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

Add

$3$ and

$1$ .

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

Step 2.3.2.6

Cancel the common factor of

$4$ and

$24$ .

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Step 2.3.2.6.1

Factor

$4$ out of

$4$ .

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

Step 2.3.2.6.2

Cancel the common factors.

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Step 2.3.2.6.2.1

Factor

$4$ out of

$24$ .

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

Step 2.3.2.6.2.2

Cancel the common factor.

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

Step 2.3.2.6.2.3

Rewrite the expression.

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

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