Calculus Examples

$f (x) = - \frac{1}{x - 1}$ at $(- 5, \frac{1}{6})$

Step 1

Find the first derivative and evaluate at $x = - 5$ and $y = \frac{1}{6}$ 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 $g (x) = \frac{1}{x - 1}$ .

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

Step 1.2

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

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

Step 1.3

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 $g (x) = x - 1$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 1.3.2

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

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

Step 1.3.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 1.4

Differentiate.

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Step 1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.4.2

Multiply ${(x - 1)}^{- 2}$ by $1$ .

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

Step 1.4.3

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 1.4.4

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

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

Step 1.4.5

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

${(x - 1)}^{- 2} (1 + 0) + \frac{1}{x - 1} \frac{d}{d x} [- 1]$

Step 1.4.6

Simplify the expression.

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Step 1.4.6.1

Add $1$ and $0$ .

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

Step 1.4.6.2

Multiply ${(x - 1)}^{- 2}$ by $1$ .

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

Step 1.4.7

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

${(x - 1)}^{- 2} + \frac{1}{x - 1} \cdot 0$

Step 1.4.8

Simplify the expression.

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Step 1.4.8.1

Multiply $\frac{1}{x - 1}$ by $0$ .

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

Step 1.4.8.2

Add ${(x - 1)}^{- 2}$ and $0$ .

${(x - 1)}^{- 2}$

Step 1.5

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

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

Step 1.6

Evaluate the derivative at $x = - 5$ .

$\frac{1}{{((- 5) - 1)}^{2}}$

Step 1.7

Simplify the denominator.

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Step 1.7.1

Subtract $1$ from $- 5$ .

$\frac{1}{{(- 6)}^{2}}$

Step 1.7.2

Raise $- 6$ to the power of $2$ .

$\frac{1}{36}$

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 $\frac{1}{36}$ and a given point $(- 5, \frac{1}{6})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{1}{6}) = \frac{1}{36} \cdot (x - (- 5))$

Step 2.2

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

$y - \frac{1}{6} = \frac{1}{36} \cdot (x + 5)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{1}{36} \cdot (x + 5)$ .

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Step 2.3.1.1

Rewrite.

$y - \frac{1}{6} = 0 + 0 + \frac{1}{36} \cdot (x + 5)$

Step 2.3.1.2

Simplify by adding zeros.

$y - \frac{1}{6} = \frac{1}{36} \cdot (x + 5)$

Step 2.3.1.3

Apply the distributive property.

$y - \frac{1}{6} = \frac{1}{36} x + \frac{1}{36} \cdot 5$

Step 2.3.1.4

Combine $\frac{1}{36}$ and $x$ .

$y - \frac{1}{6} = \frac{x}{36} + \frac{1}{36} \cdot 5$

Step 2.3.1.5

Combine $\frac{1}{36}$ and $5$ .

$y - \frac{1}{6} = \frac{x}{36} + \frac{5}{36}$

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 $\frac{1}{6}$ to both sides of the equation.

$y = \frac{x}{36} + \frac{5}{36} + \frac{1}{6}$

Step 2.3.2.2

To write $\frac{1}{6}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = \frac{x}{36} + \frac{5}{36} + \frac{1}{6} \cdot \frac{6}{6}$

Step 2.3.2.3

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.2.3.1

Multiply $\frac{1}{6}$ by $\frac{6}{6}$ .

$y = \frac{x}{36} + \frac{5}{36} + \frac{6}{6 \cdot 6}$

Step 2.3.2.3.2

Multiply $6$ by $6$ .

$y = \frac{x}{36} + \frac{5}{36} + \frac{6}{36}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{36} + \frac{5 + 6}{36}$

Step 2.3.2.5

Add $5$ and $6$ .

$y = \frac{x}{36} + \frac{11}{36}$

Step 2.3.3

Reorder terms.

$y = \frac{1}{36} x + \frac{11}{36}$

Step 3