Calculus Examples

$f (x) = \frac{1}{x - 3}$ ; $x = - 1$

Step 1

Find the corresponding $y$ -value to $x = - 1$ .

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Step 1.1

Substitute $- 1$ in for $x$ .

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

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = \frac{1}{- 1 - 3}$

Step 1.2.2

Remove parentheses.

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

Step 1.2.3

Simplify $\frac{1}{(- 1) - 3}$ .

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Step 1.2.3.1

Subtract $3$ from $- 1$ .

$y = \frac{1}{- 4}$

Step 1.2.3.2

Move the negative in front of the fraction.

$y = - \frac{1}{4}$

Step 2

Find the first derivative and evaluate at $x = - 1$ and $y = - \frac{1}{4}$ to find the slope of the tangent line.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x - 3$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $x - 3$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

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

Step 2.3.2

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

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

Step 2.3.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$- {(x - 3)}^{- 2} (1 + 0)$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

Add $1$ and $0$ .

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

Step 2.3.4.2

Multiply $- 1$ by $1$ .

$- {(x - 3)}^{- 2}$

Step 2.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.5

Evaluate the derivative at $x = - 1$ .

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

Step 2.6

Simplify the denominator.

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Step 2.6.1

Subtract $3$ from $- 1$ .

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

Step 2.6.2

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

$- \frac{1}{16}$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $- \frac{1}{16}$ and a given point $(- 1, - \frac{1}{4})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3.2

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

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite.

$y + \frac{1}{4} = 0 + 0 - \frac{1}{16} \cdot (x + 1)$

Step 3.3.1.2

Simplify by adding zeros.

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

Step 3.3.1.3

Apply the distributive property.

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

Step 3.3.1.4

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

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

Step 3.3.1.5

Multiply $- 1$ by $1$ .

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

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Subtract $\frac{1}{4}$ from both sides of the equation.

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

Step 3.3.2.2

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 3.3.2.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.3.1

Multiply $\frac{1}{4}$ by $\frac{4}{4}$ .

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

Step 3.3.2.3.2

Multiply $4$ by $4$ .

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

Step 3.3.2.4

Combine the numerators over the common denominator.

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

Step 3.3.2.5

Subtract $4$ from $- 1$ .

$y = - \frac{x}{16} + \frac{- 5}{16}$

Step 3.3.2.6

Move the negative in front of the fraction.

$y = - \frac{x}{16} - \frac{5}{16}$

Step 3.3.3

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

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Step 3.3.3.1

Reorder terms.

$y = - (\frac{1}{16} x) - \frac{5}{16}$

Step 3.3.3.2

Remove parentheses.

$y = - \frac{1}{16} x - \frac{5}{16}$

Step 4