Calculus Examples

$f (x) = \frac{x}{2 - 11 x}$ , $x = \frac{1}{11}$

Step 1

Find the corresponding $y$ -value to $x = \frac{1}{11}$ .

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Step 1.1

Substitute $\frac{1}{11}$ in for $x$ .

$y = \frac{\frac{1}{11}}{2 - 11 (\frac{1}{11})}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = \frac{\frac{1}{11}}{2 - 11 (\frac{1}{11})}$

Step 1.2.2

Simplify $\frac{\frac{1}{11}}{2 - 11 (\frac{1}{11})}$ .

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Step 1.2.2.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{1}{11} \cdot \frac{1}{2 - 11 (\frac{1}{11})}$

Step 1.2.2.2

Simplify the denominator.

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Step 1.2.2.2.1

Cancel the common factor of $11$ .

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Step 1.2.2.2.1.1

Factor $11$ out of $- 11$ .

$y = \frac{1}{11} \cdot \frac{1}{2 + 11 (- 1) \frac{1}{11}}$

Step 1.2.2.2.1.2

Cancel the common factor.

$y = \frac{1}{11} \cdot \frac{1}{2 + 11 \cdot - 1 \frac{1}{11}}$

Step 1.2.2.2.1.3

Rewrite the expression.

$y = \frac{1}{11} \cdot \frac{1}{2 - 1}$

Step 1.2.2.2.2

Subtract $1$ from $2$ .

$y = \frac{1}{11} \cdot \frac{1}{1}$

Step 1.2.2.3

Reduce the expression by cancelling the common factors.

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Step 1.2.2.3.1

Cancel the common factor of $1$ .

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Step 1.2.2.3.1.1

Cancel the common factor.

$y = \frac{1}{11} \cdot \frac{1}{1}$

Step 1.2.2.3.1.2

Rewrite the expression.

$y = \frac{1}{11} \cdot 1$

Step 1.2.2.3.2

Multiply $\frac{1}{11}$ by $1$ .

$y = \frac{1}{11}$

Step 2

Find the first derivative and evaluate at $x = \frac{1}{11}$ and $y = \frac{1}{11}$ to find the slope of the tangent line.

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Step 2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = 2 - 11 x$ .

$\frac{(2 - 11 x) \frac{d}{d x} [x] - x \frac{d}{d x} [2 - 11 x]}{{(2 - 11 x)}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

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

$\frac{(2 - 11 x) \cdot 1 - x \frac{d}{d x} [2 - 11 x]}{{(2 - 11 x)}^{2}}$

Step 2.2.2

Multiply $2 - 11 x$ by $1$ .

$\frac{2 - 11 x - x \frac{d}{d x} [2 - 11 x]}{{(2 - 11 x)}^{2}}$

Step 2.2.3

By the Sum Rule, the derivative of $2 - 11 x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- 11 x]$ .

$\frac{2 - 11 x - x (\frac{d}{d x} [2] + \frac{d}{d x} [- 11 x])}{{(2 - 11 x)}^{2}}$

Step 2.2.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{2 - 11 x - x (0 + \frac{d}{d x} [- 11 x])}{{(2 - 11 x)}^{2}}$

Step 2.2.5

Add $0$ and $\frac{d}{d x} [- 11 x]$ .

$\frac{2 - 11 x - x \frac{d}{d x} [- 11 x]}{{(2 - 11 x)}^{2}}$

Step 2.2.6

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11 x$ with respect to $x$ is $- 11 \frac{d}{d x} [x]$ .

$\frac{2 - 11 x - x (- 11 \frac{d}{d x} [x])}{{(2 - 11 x)}^{2}}$

Step 2.2.7

Multiply $- 11$ by $- 1$ .

$\frac{2 - 11 x + 11 x \frac{d}{d x} [x]}{{(2 - 11 x)}^{2}}$

Step 2.2.8

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

$\frac{2 - 11 x + 11 x \cdot 1}{{(2 - 11 x)}^{2}}$

Step 2.2.9

Simplify by adding terms.

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Step 2.2.9.1

Multiply $11$ by $1$ .

$\frac{2 - 11 x + 11 x}{{(2 - 11 x)}^{2}}$

Step 2.2.9.2

Add $- 11 x$ and $11 x$ .

$\frac{2 + 0}{{(2 - 11 x)}^{2}}$

Step 2.2.9.3

Simplify the expression.

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Step 2.2.9.3.1

Add $2$ and $0$ .

$\frac{2}{{(2 - 11 x)}^{2}}$

Step 2.2.9.3.2

Reorder terms.

$\frac{2}{{(- 11 x + 2)}^{2}}$

Step 2.3

Evaluate the derivative at $x = \frac{1}{11}$ .

$\frac{2}{{(- 11 (\frac{1}{11}) + 2)}^{2}}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the denominator.

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Step 2.4.1.1

Cancel the common factor of $11$ .

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Step 2.4.1.1.1

Factor $11$ out of $- 11$ .

$\frac{2}{{(11 (- 1) \frac{1}{11} + 2)}^{2}}$

Step 2.4.1.1.2

Cancel the common factor.

$\frac{2}{{(11 \cdot - 1 \frac{1}{11} + 2)}^{2}}$

Step 2.4.1.1.3

Rewrite the expression.

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

Step 2.4.1.2

Add $- 1$ and $2$ .

$\frac{2}{1^{2}}$

Step 2.4.1.3

One to any power is one.

$\frac{2}{1}$

Step 2.4.2

Divide $2$ by $1$ .

$2$

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 $2$ and a given point $(\frac{1}{11}, \frac{1}{11})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{1}{11}) = 2 \cdot (x - (\frac{1}{11}))$

Step 3.2

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

$y - \frac{1}{11} = 2 \cdot (x - \frac{1}{11})$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $2 \cdot (x - \frac{1}{11})$ .

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Step 3.3.1.1

Rewrite.

$y - \frac{1}{11} = 0 + 0 + 2 \cdot (x - \frac{1}{11})$

Step 3.3.1.2

Simplify by adding zeros.

$y - \frac{1}{11} = 2 \cdot (x - \frac{1}{11})$

Step 3.3.1.3

Apply the distributive property.

$y - \frac{1}{11} = 2 x + 2 (- \frac{1}{11})$

Step 3.3.1.4

Multiply $2 (- \frac{1}{11})$ .

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Step 3.3.1.4.1

Multiply $- 1$ by $2$ .

$y - \frac{1}{11} = 2 x - 2 (\frac{1}{11})$

Step 3.3.1.4.2

Combine $- 2$ and $\frac{1}{11}$ .

$y - \frac{1}{11} = 2 x + \frac{- 2}{11}$

Step 3.3.1.5

Move the negative in front of the fraction.

$y - \frac{1}{11} = 2 x - \frac{2}{11}$

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

Add $\frac{1}{11}$ to both sides of the equation.

$y = 2 x - \frac{2}{11} + \frac{1}{11}$

Step 3.3.2.2

Combine the numerators over the common denominator.

$y = 2 x + \frac{- 2 + 1}{11}$

Step 3.3.2.3

Add $- 2$ and $1$ .

$y = 2 x + \frac{- 1}{11}$

Step 3.3.2.4

Move the negative in front of the fraction.

$y = 2 x - \frac{1}{11}$

Step 4