Calculus Examples

y = \frac{1}{2 x^{3}}

$y = \frac{1}{2 x^{3}}$ ,

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

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

Step 1

Find the first derivative and evaluate at

x = 1

$x = 1$ and

y = \frac{1}{2}

$y = \frac{1}{2}$ to find the slope of the tangent line.

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Step 1.1

Since

\frac{1}{2}

$\frac{1}{2}$ is constant with respect to

x

$x$ , the derivative of

\frac{1}{2 x^{3}}

$\frac{1}{2 x^{3}}$ with respect to

x

$x$ is

\frac{1}{2} \frac{d}{d x} [\frac{1}{x^{3}}]

$\frac{1}{2} \frac{d}{d x} [\frac{1}{x^{3}}]$ .

\frac{1}{2} \frac{d}{d x} [\frac{1}{x^{3}}]

$\frac{1}{2} \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

Rewrite

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

$\frac{1}{x^{3}}$ as

{(x^{3})}^{- 1}

${(x^{3})}^{- 1}$ .

\frac{1}{2} \frac{d}{d x} [{(x^{3})}^{- 1}]

$\frac{1}{2} \frac{d}{d x} [{(x^{3})}^{- 1}]$

Step 1.2.2

Multiply the exponents in

{(x^{3})}^{- 1}

${(x^{3})}^{- 1}$ .

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Step 1.2.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

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

\frac{1}{2} \frac{d}{d x} [x^{3 \cdot - 1}]

$\frac{1}{2} \frac{d}{d x} [x^{3 \cdot - 1}]$

Step 1.2.2.2

Multiply

3

$3$ by

- 1

$- 1$ .

\frac{1}{2} \frac{d}{d x} [x^{- 3}]

$\frac{1}{2} \frac{d}{d x} [x^{- 3}]$

\frac{1}{2} \frac{d}{d x} [x^{- 3}]

$\frac{1}{2} \frac{d}{d x} [x^{- 3}]$

\frac{1}{2} \frac{d}{d x} [x^{- 3}]

$\frac{1}{2} \frac{d}{d x} [x^{- 3}]$

Step 1.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = - 3

$n = - 3$ .

\frac{1}{2} (- 3 x^{- 4})

$\frac{1}{2} (- 3 x^{- 4})$

Step 1.4

Combine fractions.

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Step 1.4.1

Combine

- 3

$- 3$ and

\frac{1}{2}

$\frac{1}{2}$ .

\frac{- 3}{2} x^{- 4}

$\frac{- 3}{2} x^{- 4}$

Step 1.4.2

Combine

\frac{- 3}{2}

$\frac{- 3}{2}$ and

x^{- 4}

$x^{- 4}$ .

\frac{- 3 x^{- 4}}{2}

$\frac{- 3 x^{- 4}}{2}$

Step 1.4.3

Simplify the expression.

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Step 1.4.3.1

Move

x^{- 4}

$x^{- 4}$ to the denominator using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

\frac{- 3}{2 x^{4}}

$\frac{- 3}{2 x^{4}}$

Step 1.4.3.2

Move the negative in front of the fraction.

- \frac{3}{2 x^{4}}

$- \frac{3}{2 x^{4}}$

- \frac{3}{2 x^{4}}

$- \frac{3}{2 x^{4}}$

- \frac{3}{2 x^{4}}

$- \frac{3}{2 x^{4}}$

Step 1.5

Evaluate the derivative at

x = 1

$x = 1$ .

- \frac{3}{2 {(1)}^{4}}

$- \frac{3}{2 {(1)}^{4}}$

Step 1.6

Simplify.

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Step 1.6.1

One to any power is one.

- \frac{3}{2 \cdot 1}

$- \frac{3}{2 \cdot 1}$

Step 1.6.2

Multiply

2

$2$ by

1

$1$ .

- \frac{3}{2}

$- \frac{3}{2}$

- \frac{3}{2}

$- \frac{3}{2}$

- \frac{3}{2}

$- \frac{3}{2}$

Step 2

Plug the slope and point values into the point-slope formula and solve for

y

$y$ .

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Step 2.1

Use the slope

- \frac{3}{2}

$- \frac{3}{2}$ and a given point

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

$(1, \frac{1}{2})$ to substitute for

x_{1}

$x_{1}$ and

y_{1}

$y_{1}$ in the point-slope form

y - y_{1} = m (x - x_{1})

$y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

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

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

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$- \frac{3}{2} \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Combine

$x$ and

$\frac{3}{2}$ .

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

Step 2.3.1.5

Multiply

$- \frac{3}{2} \cdot - 1$ .

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Step 2.3.1.5.1

Multiply

$- 1$ by

$- 1$ .

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

Step 2.3.1.5.2

Multiply

$\frac{3}{2}$ by

$1$ .

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

Step 2.3.1.6

Move

$3$ to the left of

$x$ .

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

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

$\frac{1}{2}$ to both sides of the equation.

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

Step 2.3.2.2

Combine the numerators over the common denominator.

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

Step 2.3.2.3

Add

$3$ and

$1$ .

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

Step 2.3.2.4

Split the fraction

$\frac{- 3 x + 4}{2}$ into two fractions.

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

Step 2.3.2.5

Simplify each term.

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Step 2.3.2.5.1

Move the negative in front of the fraction.

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

Step 2.3.2.5.2

Divide

$4$ by

$2$ .

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

Step 2.3.3

Write in

$y = m x + b$ form.

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Step 2.3.3.1

Reorder terms.

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

Step 2.3.3.2

Remove parentheses.

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

Step 3