Calculus Examples

$x^{2} y^{2} - 9 x^{2} - 4 y^{2} = 0$ , $(4, - 2 \sqrt{3})$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = - 2 \sqrt{3}$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} y^{2} - 9 x^{2} - 4 y^{2}) = \frac{d}{d x} (0)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{2} y^{2} - 9 x^{2} - 4 y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2} y^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$ .

$\frac{d}{d x} [x^{2} y^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [x^{2} y^{2}]$ .

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Step 1.2.2.1

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 $f (x) = x^{2}$ and $g (x) = y^{2}$ .

$x^{2} \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 1.2.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 1.2.2.2.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

$x^{2} (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 1.2.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$x^{2} (2 u_{1} \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 1.2.2.2.3

Replace all occurrences of $u_{1}$ with $y$ .

$x^{2} (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 1.2.2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$x^{2} (2 y y') + y^{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 1.2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{2} (2 y y') + y^{2} (2 x) + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 1.2.2.5

Move $2$ to the left of $x^{2}$ .

$2 \cdot x^{2} y y' + y^{2} (2 x) + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 1.2.2.6

Move $2$ to the left of $y^{2}$ .

$2 x^{2} y y' + 2 y^{2} x + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 9 x^{2}]$ .

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Step 1.2.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{2}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{2}]$ .

$2 x^{2} y y' + 2 y^{2} x - 9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 1.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x^{2} y y' + 2 y^{2} x - 9 (2 x) + \frac{d}{d x} [- 4 y^{2}]$

Step 1.2.3.3

Multiply $2$ by $- 9$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x + \frac{d}{d x} [- 4 y^{2}]$

Step 1.2.4

Evaluate $\frac{d}{d x} [- 4 y^{2}]$ .

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Step 1.2.4.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 y^{2}$ with respect to $x$ is $- 4 \frac{d}{d x} [y^{2}]$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x - 4 \frac{d}{d x} [y^{2}]$

Step 1.2.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 1.2.4.2.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x - 4 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y])$

Step 1.2.4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x - 4 (2 u_{2} \frac{d}{d x} [y])$

Step 1.2.4.2.3

Replace all occurrences of $u_{2}$ with $y$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x - 4 (2 y \frac{d}{d x} [y])$

Step 1.2.4.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x - 4 (2 y y')$

Step 1.2.4.4

Multiply $2$ by $- 4$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x - 8 y y'$

Step 1.2.5

Reorder terms.

$2 x^{2} y y' + 2 y^{2} x - 8 y y' - 18 x$

Step 1.3

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

$0$

Step 1.4

Reform the equation by setting the left side equal to the right side.

$2 x^{2} y y' + 2 y^{2} x - 8 y y' - 18 x = 0$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

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

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Step 1.5.1.1

Subtract $2 y^{2} x$ from both sides of the equation.

$2 x^{2} y y' - 8 y y' - 18 x = - 2 y^{2} x$

Step 1.5.1.2

Add $18 x$ to both sides of the equation.

$2 x^{2} y y' - 8 y y' = - 2 y^{2} x + 18 x$

Step 1.5.2

Factor $2 y y'$ out of $2 x^{2} y y' - 8 y y'$ .

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Step 1.5.2.1

Factor $2 y y'$ out of $2 x^{2} y y'$ .

$2 y y' x^{2} - 8 y y' = - 2 y^{2} x + 18 x$

Step 1.5.2.2

Factor $2 y y'$ out of $- 8 y y'$ .

$2 y y' x^{2} + 2 y y' \cdot - 4 = - 2 y^{2} x + 18 x$

Step 1.5.2.3

Factor $2 y y'$ out of $2 y y' x^{2} + 2 y y' \cdot - 4$ .

$2 y y' (x^{2} - 4) = - 2 y^{2} x + 18 x$

Step 1.5.3

Rewrite $4$ as $2^{2}$ .

$2 y y' (x^{2} - 2^{2}) = - 2 y^{2} x + 18 x$

Step 1.5.4

Factor.

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Step 1.5.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$2 y y' ((x + 2) (x - 2)) = - 2 y^{2} x + 18 x$

Step 1.5.4.2

Remove unnecessary parentheses.

$2 y y' (x + 2) (x - 2) = - 2 y^{2} x + 18 x$

Step 1.5.5

Divide each term in $2 y y' (x + 2) (x - 2) = - 2 y^{2} x + 18 x$ by $2 y (x + 2) (x - 2)$ and simplify.

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Step 1.5.5.1

Divide each term in $2 y y' (x + 2) (x - 2) = - 2 y^{2} x + 18 x$ by $2 y (x + 2) (x - 2)$ .

$\frac{2 y y' (x + 2) (x - 2)}{2 y (x + 2) (x - 2)} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.2

Simplify the left side.

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Step 1.5.5.2.1

Cancel the common factor of $2$ .

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Step 1.5.5.2.1.1

Cancel the common factor.

$\frac{2 y y' (x + 2) (x - 2)}{2 y (x + 2) (x - 2)} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.2.1.2

Rewrite the expression.

$\frac{y y' (x + 2) (x - 2)}{(y (x + 2)) (x - 2)} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.2.2

Cancel the common factor of $y$ .

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Step 1.5.5.2.2.1

Cancel the common factor.

$\frac{y y' (x + 2) (x - 2)}{y (x + 2) (x - 2)} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.2.2.2

Rewrite the expression.

$\frac{y' (x + 2) (x - 2)}{(x + 2) (x - 2)} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.2.3

Cancel the common factor of $x + 2$ .

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Step 1.5.5.2.3.1

Cancel the common factor.

$\frac{y' (x + 2) (x - 2)}{(x + 2) (x - 2)} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.2.3.2

Rewrite the expression.

$\frac{y' (x - 2)}{x - 2} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.2.4

Cancel the common factor of $x - 2$ .

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Step 1.5.5.2.4.1

Cancel the common factor.

$\frac{y' (x - 2)}{x - 2} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.2.4.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.3

Simplify the right side.

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Step 1.5.5.3.1

Simplify each term.

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Step 1.5.5.3.1.1

Cancel the common factor of $- 2$ and $2$ .

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Step 1.5.5.3.1.1.1

Factor $2$ out of $- 2 y^{2} x$ .

$y' = \frac{2 (- y^{2} x)}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.3.1.1.2

Cancel the common factors.

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Step 1.5.5.3.1.1.2.1

Factor $2$ out of $2 y (x + 2) (x - 2)$ .

$y' = \frac{2 (- y^{2} x)}{2 ((y (x + 2)) (x - 2))} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.3.1.1.2.2

Cancel the common factor.

$y' = \frac{2 (- y^{2} x)}{2 ((y (x + 2)) (x - 2))} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- y^{2} x}{(y (x + 2)) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.3.1.2

Cancel the common factor of $y^{2}$ and $y$ .

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Step 1.5.5.3.1.2.1

Factor $y$ out of $- y^{2} x$ .

$y' = \frac{y (- y x)}{(y (x + 2)) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.3.1.2.2

Cancel the common factors.

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Step 1.5.5.3.1.2.2.1

Factor $y$ out of $(y (x + 2)) (x - 2)$ .

$y' = \frac{y (- y x)}{y ((x + 2) (x - 2))} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.3.1.2.2.2

Cancel the common factor.

$y' = \frac{y (- y x)}{y ((x + 2) (x - 2))} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.3.1.2.2.3

Rewrite the expression.

$y' = \frac{- y x}{(x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.3.1.3

Move the negative in front of the fraction.

$y' = - \frac{y x}{(x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 1.5.5.3.1.4

Cancel the common factor of $18$ and $2$ .

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Step 1.5.5.3.1.4.1

Factor $2$ out of $18 x$ .

$y' = - \frac{y x}{(x + 2) (x - 2)} + \frac{2 (9 x)}{2 y (x + 2) (x - 2)}$

Step 1.5.5.3.1.4.2

Cancel the common factors.

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Step 1.5.5.3.1.4.2.1

Factor $2$ out of $2 y (x + 2) (x - 2)$ .

$y' = - \frac{y x}{(x + 2) (x - 2)} + \frac{2 (9 x)}{2 ((y (x + 2)) (x - 2))}$

Step 1.5.5.3.1.4.2.2

Cancel the common factor.

$y' = - \frac{y x}{(x + 2) (x - 2)} + \frac{2 (9 x)}{2 ((y (x + 2)) (x - 2))}$

Step 1.5.5.3.1.4.2.3

Rewrite the expression.

$y' = - \frac{y x}{(x + 2) (x - 2)} + \frac{9 x}{(y (x + 2)) (x - 2)}$

$y' = - \frac{y x}{(x + 2) (x - 2)} + \frac{9 x}{y (x + 2) (x - 2)}$

Step 1.6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{y x}{(x + 2) (x - 2)} + \frac{9 x}{y (x + 2) (x - 2)}$

Step 1.7

Evaluate at $x = 4$ and $y = - 2 \sqrt{3}$ .

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Step 1.7.1

Replace the variable $x$ with $4$ in the expression.

$- \frac{y (4)}{((4) + 2) ((4) - 2)} + \frac{9 (4)}{y ((4) + 2) ((4) - 2)}$

Step 1.7.2

Replace the variable $y$ with $- 2 \sqrt{3}$ in the expression.

$- \frac{(- 2 \sqrt{3}) (4)}{((4) + 2) ((4) - 2)} + \frac{9 (4)}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.3

Multiply $9$ by $4$ .

$- \frac{(- 2 \sqrt{3}) (4)}{((4) + 2) ((4) - 2)} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4

Simplify each term.

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Step 1.7.4.1

Cancel the common factor of $- 2$ and $(4) + 2$ .

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Step 1.7.4.1.1

Factor $2$ out of $(- 2 \sqrt{3}) (4)$ .

$- \frac{2 (- \sqrt{3} \cdot 4)}{((4) + 2) ((4) - 2)} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.1.2

Cancel the common factors.

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Step 1.7.4.1.2.1

Factor $2$ out of $((4) + 2) ((4) - 2)$ .

$- \frac{2 (- \sqrt{3} \cdot 4)}{2 ((2 + 1) ((4) - 2))} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.1.2.2

Cancel the common factor.

$- \frac{2 (- \sqrt{3} \cdot 4)}{2 ((2 + 1) ((4) - 2))} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.1.2.3

Rewrite the expression.

$- \frac{- \sqrt{3} \cdot 4}{(2 + 1) ((4) - 2)} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.2

Cancel the common factor of $4$ and $(4) - 2$ .

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Step 1.7.4.2.1

Factor $2$ out of $- \sqrt{3} \cdot 4$ .

$- \frac{2 (- \sqrt{3} \cdot 2)}{(2 + 1) ((4) - 2)} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.2.2

Cancel the common factors.

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Step 1.7.4.2.2.1

Factor $2$ out of $(2 + 1) ((4) - 2)$ .

$- \frac{2 (- \sqrt{3} \cdot 2)}{2 ((2 + 1) (2 - 1))} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.2.2.2

Cancel the common factor.

$- \frac{2 (- \sqrt{3} \cdot 2)}{2 ((2 + 1) (2 - 1))} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.2.2.3

Rewrite the expression.

$- \frac{- \sqrt{3} \cdot 2}{(2 + 1) (2 - 1)} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.3

Multiply $2$ by $- 1$ .

$- \frac{- 2 \sqrt{3}}{(2 + 1) (2 - 1)} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.4

Simplify the denominator.

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Step 1.7.4.4.1

Add $2$ and $1$ .

$- \frac{- 2 \sqrt{3}}{3 (2 - 1)} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.4.2

Subtract $1$ from $2$ .

$- \frac{- 2 \sqrt{3}}{3 \cdot 1} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.5

Multiply $3$ by $1$ .

$- \frac{- 2 \sqrt{3}}{3} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.6

Move the negative in front of the fraction.

$- - \frac{2 \sqrt{3}}{3} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.7

Multiply $- - \frac{2 \sqrt{3}}{3}$ .

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Step 1.7.4.7.1

Multiply $- 1$ by $- 1$ .

$1 \frac{2 \sqrt{3}}{3} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.7.2

Multiply $\frac{2 \sqrt{3}}{3}$ by $1$ .

$\frac{2 \sqrt{3}}{3} + \frac{36}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.8

Cancel the common factor of $36$ and $- 2$ .

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Step 1.7.4.8.1

Factor $2$ out of $36$ .

$\frac{2 \sqrt{3}}{3} + \frac{2 (18)}{(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)}$

Step 1.7.4.8.2

Cancel the common factors.

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Step 1.7.4.8.2.1

Factor $2$ out of $(- 2 \sqrt{3}) ((4) + 2) ((4) - 2)$ .

$\frac{2 \sqrt{3}}{3} + \frac{2 (18)}{2 (- \sqrt{3} ((4) + 2) ((4) - 2))}$

Step 1.7.4.8.2.2

Cancel the common factor.

$\frac{2 \sqrt{3}}{3} + \frac{2 \cdot 18}{2 (- \sqrt{3} ((4) + 2) ((4) - 2))}$

Step 1.7.4.8.2.3

Rewrite the expression.

$\frac{2 \sqrt{3}}{3} + \frac{18}{- \sqrt{3} ((4) + 2) ((4) - 2)}$

Step 1.7.4.9

Cancel the common factor of $18$ and $(4) + 2$ .

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Step 1.7.4.9.1

Factor $2$ out of $18$ .

$\frac{2 \sqrt{3}}{3} + \frac{2 \cdot 9}{- \sqrt{3} (4 + 2) ((4) - 2)}$

Step 1.7.4.9.2

Cancel the common factors.

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Step 1.7.4.9.2.1

Factor $2$ out of $- \sqrt{3} (4 + 2) ((4) - 2)$ .

$\frac{2 \sqrt{3}}{3} + \frac{2 \cdot 9}{2 (- \sqrt{3} (2 + 1) ((4) - 2))}$

Step 1.7.4.9.2.2

Cancel the common factor.

$\frac{2 \sqrt{3}}{3} + \frac{2 \cdot 9}{2 (- \sqrt{3} (2 + 1) ((4) - 2))}$

Step 1.7.4.9.2.3

Rewrite the expression.

$\frac{2 \sqrt{3}}{3} + \frac{9}{- \sqrt{3} (2 + 1) ((4) - 2)}$

Step 1.7.4.10

Simplify the denominator.

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Step 1.7.4.10.1

Add $2$ and $1$ .

$\frac{2 \sqrt{3}}{3} + \frac{9}{- \sqrt{3} \cdot 3 (4 - 2)}$

Step 1.7.4.10.2

Multiply $3$ by $- 1$ .

$\frac{2 \sqrt{3}}{3} + \frac{9}{- 3 \sqrt{3} (4 - 2)}$

Step 1.7.4.10.3

Subtract $2$ from $4$ .

$\frac{2 \sqrt{3}}{3} + \frac{9}{- 3 \sqrt{3} \cdot 2}$

Step 1.7.4.10.4

Multiply $2$ by $- 3$ .

$\frac{2 \sqrt{3}}{3} + \frac{9}{- 6 \sqrt{3}}$

Step 1.7.4.11

Cancel the common factor of $9$ and $- 6$ .

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Step 1.7.4.11.1

Factor $3$ out of $9$ .

$\frac{2 \sqrt{3}}{3} + \frac{3 (3)}{- 6 \sqrt{3}}$

Step 1.7.4.11.2

Cancel the common factors.

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Step 1.7.4.11.2.1

Factor $3$ out of $- 6 \sqrt{3}$ .

$\frac{2 \sqrt{3}}{3} + \frac{3 (3)}{3 (- 2 \sqrt{3})}$

Step 1.7.4.11.2.2

Cancel the common factor.

$\frac{2 \sqrt{3}}{3} + \frac{3 \cdot 3}{3 (- 2 \sqrt{3})}$

Step 1.7.4.11.2.3

Rewrite the expression.

$\frac{2 \sqrt{3}}{3} + \frac{3}{- 2 \sqrt{3}}$

Step 1.7.4.12

Move the negative in front of the fraction.

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

Step 1.7.4.13

Multiply $\frac{3}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{2 \sqrt{3}}{3} - (\frac{3}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 1.7.4.14

Combine and simplify the denominator.

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Step 1.7.4.14.1

Multiply $\frac{3}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.7.4.14.2

Move $\sqrt{3}$ .

$\frac{2 \sqrt{3}}{3} - \frac{3 \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

Step 1.7.4.14.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 1.7.4.14.4

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 1.7.4.14.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sqrt{3}}{3} - \frac{3 \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}}$

Step 1.7.4.14.6

Add $1$ and $1$ .

$\frac{2 \sqrt{3}}{3} - \frac{3 \sqrt{3}}{2 {\sqrt{3}}^{2}}$

Step 1.7.4.14.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.7.4.14.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 1.7.4.14.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.7.4.14.7.3

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

$\frac{2 \sqrt{3}}{3} - \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 1.7.4.14.7.4

Cancel the common factor of $2$ .

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Step 1.7.4.14.7.4.1

Cancel the common factor.

$\frac{2 \sqrt{3}}{3} - \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 1.7.4.14.7.4.2

Rewrite the expression.

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

Step 1.7.4.14.7.5

Evaluate the exponent.

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

Step 1.7.4.15

Cancel the common factor of $3$ .

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Step 1.7.4.15.1

Cancel the common factor.

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

Step 1.7.4.15.2

Rewrite the expression.

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

Step 1.7.5

To write $\frac{2 \sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.7.6

To write $- \frac{\sqrt{3}}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.7.7

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.7.7.1

Multiply $\frac{2 \sqrt{3}}{3}$ by $\frac{2}{2}$ .

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

Step 1.7.7.2

Multiply $3$ by $2$ .

$\frac{2 \sqrt{3} \cdot 2}{6} - \frac{\sqrt{3}}{2} \cdot \frac{3}{3}$

Step 1.7.7.3

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{3}{3}$ .

$\frac{2 \sqrt{3} \cdot 2}{6} - \frac{\sqrt{3} \cdot 3}{2 \cdot 3}$

Step 1.7.7.4

Multiply $2$ by $3$ .

$\frac{2 \sqrt{3} \cdot 2}{6} - \frac{\sqrt{3} \cdot 3}{6}$

Step 1.7.8

Combine the numerators over the common denominator.

$\frac{2 \sqrt{3} \cdot 2 - \sqrt{3} \cdot 3}{6}$

Step 1.7.9

Simplify the numerator.

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Step 1.7.9.1

Multiply $2$ by $2$ .

$\frac{4 \sqrt{3} - \sqrt{3} \cdot 3}{6}$

Step 1.7.9.2

Multiply $3$ by $- 1$ .

$\frac{4 \sqrt{3} - 3 \sqrt{3}}{6}$

Step 1.7.9.3

Subtract $3 \sqrt{3}$ from $4 \sqrt{3}$ .

$\frac{\sqrt{3}}{6}$

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 $\frac{\sqrt{3}}{6}$ and a given point $(4, - 2 \sqrt{3})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2 \sqrt{3}) = \frac{\sqrt{3}}{6} \cdot (x - (4))$

Step 2.2

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

$y + 2 \sqrt{3} = \frac{\sqrt{3}}{6} \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{\sqrt{3}}{6} \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y + 2 \sqrt{3} = 0 + 0 + \frac{\sqrt{3}}{6} \cdot (x - 4)$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

$y + 2 \sqrt{3} = \frac{\sqrt{3}}{6} x + \frac{\sqrt{3}}{6} \cdot - 4$

Step 2.3.1.2.2

Combine $\frac{\sqrt{3}}{6}$ and $x$ .

$y + 2 \sqrt{3} = \frac{\sqrt{3} x}{6} + \frac{\sqrt{3}}{6} \cdot - 4$

Step 2.3.1.2.3

Cancel the common factor of $2$ .

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Step 2.3.1.2.3.1

Factor $2$ out of $6$ .

$y + 2 \sqrt{3} = \frac{\sqrt{3} x}{6} + \frac{\sqrt{3}}{2 (3)} \cdot - 4$

Step 2.3.1.2.3.2

Factor $2$ out of $- 4$ .

$y + 2 \sqrt{3} = \frac{\sqrt{3} x}{6} + \frac{\sqrt{3}}{2 \cdot 3} \cdot (2 \cdot - 2)$

Step 2.3.1.2.3.3

Cancel the common factor.

$y + 2 \sqrt{3} = \frac{\sqrt{3} x}{6} + \frac{\sqrt{3}}{2 \cdot 3} \cdot (2 \cdot - 2)$

Step 2.3.1.2.3.4

Rewrite the expression.

$y + 2 \sqrt{3} = \frac{\sqrt{3} x}{6} + \frac{\sqrt{3}}{3} \cdot - 2$

Step 2.3.1.2.4

Combine $\frac{\sqrt{3}}{3}$ and $- 2$ .

$y + 2 \sqrt{3} = \frac{\sqrt{3} x}{6} + \frac{\sqrt{3} \cdot - 2}{3}$

Step 2.3.1.3

Simplify each term.

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Step 2.3.1.3.1

Move $- 2$ to the left of $\sqrt{3}$ .

$y + 2 \sqrt{3} = \frac{\sqrt{3} x}{6} + \frac{- 2 \sqrt{3}}{3}$

Step 2.3.1.3.2

Move the negative in front of the fraction.

$y + 2 \sqrt{3} = \frac{\sqrt{3} x}{6} - \frac{2 \sqrt{3}}{3}$

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

Subtract $2 \sqrt{3}$ from both sides of the equation.

$y = \frac{\sqrt{3} x}{6} - \frac{2 \sqrt{3}}{3} - 2 \sqrt{3}$

Step 2.3.2.2

To write $- 2 \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{\sqrt{3} x}{6} - \frac{2 \sqrt{3}}{3} - 2 \sqrt{3} \cdot \frac{3}{3}$

Step 2.3.2.3

Combine $- 2 \sqrt{3}$ and $\frac{3}{3}$ .

$y = \frac{\sqrt{3} x}{6} - \frac{2 \sqrt{3}}{3} + \frac{- 2 \sqrt{3} \cdot 3}{3}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{\sqrt{3} x}{6} + \frac{- 2 \sqrt{3} - 2 \sqrt{3} \cdot 3}{3}$

Step 2.3.2.5

Simplify each term.

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Step 2.3.2.5.1

Simplify the numerator.

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Step 2.3.2.5.1.1

Multiply $3$ by $- 2$ .

$y = \frac{\sqrt{3} x}{6} + \frac{- 2 \sqrt{3} - 6 \sqrt{3}}{3}$

Step 2.3.2.5.1.2

Subtract $6 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$y = \frac{\sqrt{3} x}{6} + \frac{- 8 \sqrt{3}}{3}$

Step 2.3.2.5.2

Move the negative in front of the fraction.

$y = \frac{\sqrt{3} x}{6} - \frac{8 \sqrt{3}}{3}$

Step 2.3.3

Reorder terms.

$y = \frac{\sqrt{3}}{6} x - \frac{8 \sqrt{3}}{3}$

Step 3