Calculus Examples

$3 x + y^{3} - \frac{4 y}{x + 2} = 10 x^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 x + y^{3} - \frac{4 y}{x + 2}) = \frac{d}{d x} (10 x^{2})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $3 x + y^{3} - \frac{4 y}{x + 2}$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [y^{3}] + \frac{d}{d x} [- \frac{4 y}{x + 2}]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [y^{3}] + \frac{d}{d x} [- \frac{4 y}{x + 2}]$

Step 2.2

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x] + \frac{d}{d x} [y^{3}] + \frac{d}{d x} [- \frac{4 y}{x + 2}]$

Step 2.2.2

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

$3 \cdot 1 + \frac{d}{d x} [y^{3}] + \frac{d}{d x} [- \frac{4 y}{x + 2}]$

Step 2.2.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [y^{3}] + \frac{d}{d x} [- \frac{4 y}{x + 2}]$

Step 2.3

Evaluate $\frac{d}{d x} [y^{3}]$ .

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Step 2.3.1

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

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Step 2.3.1.1

To apply the Chain Rule, set $u$ as $y$ .

$3 + \frac{d}{d u} [u^{3}] \frac{d}{d x} [y] + \frac{d}{d x} [- \frac{4 y}{x + 2}]$

Step 2.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 + 3 u^{2} \frac{d}{d x} [y] + \frac{d}{d x} [- \frac{4 y}{x + 2}]$

Step 2.3.1.3

Replace all occurrences of $u$ with $y$ .

$3 + 3 y^{2} \frac{d}{d x} [y] + \frac{d}{d x} [- \frac{4 y}{x + 2}]$

Step 2.3.2

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

$3 + 3 y^{2} y' + \frac{d}{d x} [- \frac{4 y}{x + 2}]$

Step 2.4

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

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Step 2.4.1

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

$3 + 3 y^{2} y' - 4 \frac{d}{d x} [\frac{y}{x + 2}]$

Step 2.4.2

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 $f (x) = y$ and $g (x) = x + 2$ .

$3 + 3 y^{2} y' - 4 \frac{(x + 2) \frac{d}{d x} [y] - y \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 2.4.3

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

$3 + 3 y^{2} y' - 4 \frac{(x + 2) y' - y \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 2.4.4

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

$3 + 3 y^{2} y' - 4 \frac{(x + 2) y' - y (\frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 2.4.5

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

$3 + 3 y^{2} y' - 4 \frac{(x + 2) y' - y (1 + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 2.4.6

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

$3 + 3 y^{2} y' - 4 \frac{(x + 2) y' - y (1 + 0)}{{(x + 2)}^{2}}$

Step 2.4.7

Add $1$ and $0$ .

$3 + 3 y^{2} y' - 4 \frac{(x + 2) y' - y \cdot 1}{{(x + 2)}^{2}}$

Step 2.4.8

Multiply $- 1$ by $1$ .

$3 + 3 y^{2} y' - 4 \frac{(x + 2) y' - y}{{(x + 2)}^{2}}$

Step 2.4.9

Combine $- 4$ and $\frac{(x + 2) y' - y}{{(x + 2)}^{2}}$ .

$3 + 3 y^{2} y' + \frac{- 4 ((x + 2) y' - y)}{{(x + 2)}^{2}}$

Step 2.4.10

Move the negative in front of the fraction.

$3 + 3 y^{2} y' - \frac{4 ((x + 2) y' - y)}{{(x + 2)}^{2}}$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$3 + 3 y^{2} y' - \frac{4 (x y' + 2 y' - y)}{{(x + 2)}^{2}}$

Step 2.5.2

Apply the distributive property.

$3 + 3 y^{2} y' - \frac{4 (x y') + 4 (2 y') + 4 (- y)}{{(x + 2)}^{2}}$

Step 2.5.3

Combine terms.

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Step 2.5.3.1

Multiply $2$ by $4$ .

$3 + 3 y^{2} y' - \frac{4 x y' + 8 y' + 4 (- y)}{{(x + 2)}^{2}}$

Step 2.5.3.2

Multiply $- 1$ by $4$ .

$3 + 3 y^{2} y' - \frac{4 x y' + 8 y' - 4 y}{{(x + 2)}^{2}}$

Step 2.5.3.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{{(x + 2)}^{2}}{{(x + 2)}^{2}}$ .

$3 y^{2} y' + \frac{3 {(x + 2)}^{2}}{{(x + 2)}^{2}} - \frac{4 x y' + 8 y' - 4 y}{{(x + 2)}^{2}}$

Step 2.5.3.4

Combine the numerators over the common denominator.

$3 y^{2} y' + \frac{3 {(x + 2)}^{2} - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}}$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{2}$ with respect to $x$ is $10 \frac{d}{d x} [x^{2}]$ .

$10 \frac{d}{d x} [x^{2}]$

Step 3.2

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

$10 (2 x)$

Step 3.3

Multiply $2$ by $10$ .

$20 x$

Step 4

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

$3 y^{2} y' + \frac{3 {(x + 2)}^{2} - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5

Solve for $y'$ .

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Step 5.1

Factor each term.

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Step 5.1.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$3 y^{2} y' + \frac{3 ((x + 2) (x + 2)) - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 5.1.2.1

Apply the distributive property.

$3 y^{2} y' + \frac{3 (x (x + 2) + 2 (x + 2)) - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.2.2

Apply the distributive property.

$3 y^{2} y' + \frac{3 (x \cdot x + x \cdot 2 + 2 (x + 2)) - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.2.3

Apply the distributive property.

$3 y^{2} y' + \frac{3 (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.3

Simplify and combine like terms.

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Step 5.1.3.1

Simplify each term.

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Step 5.1.3.1.1

Multiply $x$ by $x$ .

$3 y^{2} y' + \frac{3 (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.3.1.2

Move $2$ to the left of $x$ .

$3 y^{2} y' + \frac{3 (x^{2} + 2 x + 2 x + 2 \cdot 2) - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.3.1.3

Multiply $2$ by $2$ .

$3 y^{2} y' + \frac{3 (x^{2} + 2 x + 2 x + 4) - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.3.2

Add $2 x$ and $2 x$ .

$3 y^{2} y' + \frac{3 (x^{2} + 4 x + 4) - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.4

Apply the distributive property.

$3 y^{2} y' + \frac{3 x^{2} + 3 (4 x) + 3 \cdot 4 - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.5

Simplify.

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Step 5.1.5.1

Multiply $4$ by $3$ .

$3 y^{2} y' + \frac{3 x^{2} + 12 x + 3 \cdot 4 - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.5.2

Multiply $3$ by $4$ .

$3 y^{2} y' + \frac{3 x^{2} + 12 x + 12 - (4 x y' + 8 y' - 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.6

Apply the distributive property.

$3 y^{2} y' + \frac{3 x^{2} + 12 x + 12 - (4 x y') - (8 y') - (- 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.7

Simplify.

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Step 5.1.7.1

Multiply $4$ by $- 1$ .

$3 y^{2} y' + \frac{3 x^{2} + 12 x + 12 - 4 (x y') - (8 y') - (- 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.7.2

Multiply $8$ by $- 1$ .

$3 y^{2} y' + \frac{3 x^{2} + 12 x + 12 - 4 (x y') - 8 y' - (- 4 y)}{{(x + 2)}^{2}} = 20 x$

Step 5.1.7.3

Multiply $- 4$ by $- 1$ .

$3 y^{2} y' + \frac{3 x^{2} + 12 x + 12 - 4 (x y') - 8 y' + 4 y}{{(x + 2)}^{2}} = 20 x$

Step 5.1.8

Remove parentheses.

$3 y^{2} y' + \frac{3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y}{{(x + 2)}^{2}} = 20 x$

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, {(x + 2)}^{2}, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

${(x + 2)}^{2}$

Step 5.3

Multiply each term in $3 y^{2} y' + \frac{3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y}{{(x + 2)}^{2}} = 20 x$ by ${(x + 2)}^{2}$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $3 y^{2} y' + \frac{3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y}{{(x + 2)}^{2}} = 20 x$ by ${(x + 2)}^{2}$ .

$3 y^{2} y' {(x + 2)}^{2} + \frac{3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y}{{(x + 2)}^{2}} {(x + 2)}^{2} = 20 x {(x + 2)}^{2}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of ${(x + 2)}^{2}$ .

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Step 5.3.2.1.1

Cancel the common factor.

$3 y^{2} y' {(x + 2)}^{2} + \frac{3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y}{{(x + 2)}^{2}} {(x + 2)}^{2} = 20 x {(x + 2)}^{2}$

Step 5.3.2.1.2

Rewrite the expression.

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x {(x + 2)}^{2}$

Step 5.4

Solve the equation.

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Step 5.4.1

Simplify $20 x {(x + 2)}^{2}$ .

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Step 5.4.1.1

Rewrite.

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 0 + 0 + 20 x {(x + 2)}^{2}$

Step 5.4.1.2

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x ((x + 2) (x + 2))$

Step 5.4.1.3

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 5.4.1.3.1

Apply the distributive property.

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x (x (x + 2) + 2 (x + 2))$

Step 5.4.1.3.2

Apply the distributive property.

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x (x \cdot x + x \cdot 2 + 2 (x + 2))$

Step 5.4.1.3.3

Apply the distributive property.

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2)$

Step 5.4.1.4

Simplify and combine like terms.

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Step 5.4.1.4.1

Simplify each term.

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Step 5.4.1.4.1.1

Multiply $x$ by $x$ .

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2)$

Step 5.4.1.4.1.2

Move $2$ to the left of $x$ .

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2)$

Step 5.4.1.4.1.3

Multiply $2$ by $2$ .

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x (x^{2} + 2 x + 2 x + 4)$

Step 5.4.1.4.2

Add $2 x$ and $2 x$ .

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x (x^{2} + 4 x + 4)$

Step 5.4.1.5

Apply the distributive property.

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x \cdot x^{2} + 20 x (4 x) + 20 x \cdot 4$

Step 5.4.1.6

Simplify.

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Step 5.4.1.6.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.4.1.6.1.1

Move $x^{2}$ .

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 (x^{2} x) + 20 x (4 x) + 20 x \cdot 4$

Step 5.4.1.6.1.2

Multiply $x^{2}$ by $x$ .

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Step 5.4.1.6.1.2.1

Raise $x$ to the power of $1$ .

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 (x^{2} x^{1}) + 20 x (4 x) + 20 x \cdot 4$

Step 5.4.1.6.1.2.2

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

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x^{2 + 1} + 20 x (4 x) + 20 x \cdot 4$

Step 5.4.1.6.1.3

Add $2$ and $1$ .

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x^{3} + 20 x (4 x) + 20 x \cdot 4$

Step 5.4.1.6.2

Rewrite using the commutative property of multiplication.

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x^{3} + 20 \cdot 4 x \cdot x + 20 x \cdot 4$

Step 5.4.1.6.3

Multiply $4$ by $20$ .

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x^{3} + 20 \cdot 4 x \cdot x + 80 x$

Step 5.4.1.7

Simplify each term.

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Step 5.4.1.7.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.4.1.7.1.1

Move $x$ .

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x^{3} + 20 \cdot 4 (x \cdot x) + 80 x$

Step 5.4.1.7.1.2

Multiply $x$ by $x$ .

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x^{3} + 20 \cdot 4 x^{2} + 80 x$

Step 5.4.1.7.2

Multiply $20$ by $4$ .

$3 y^{2} y' {(x + 2)}^{2} + 3 x^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x^{3} + 80 x^{2} + 80 x$

Step 5.4.2

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

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Step 5.4.2.1

Subtract $3 x^{2}$ from both sides of the equation.

$3 y^{2} y' {(x + 2)}^{2} + 12 x + 12 - 4 x y' - 8 y' + 4 y = 20 x^{3} + 80 x^{2} + 80 x - 3 x^{2}$

Step 5.4.2.2

Subtract $12 x$ from both sides of the equation.

$3 y^{2} y' {(x + 2)}^{2} + 12 - 4 x y' - 8 y' + 4 y = 20 x^{3} + 80 x^{2} + 80 x - 3 x^{2} - 12 x$

Step 5.4.2.3

Subtract $12$ from both sides of the equation.

$3 y^{2} y' {(x + 2)}^{2} - 4 x y' - 8 y' + 4 y = 20 x^{3} + 80 x^{2} + 80 x - 3 x^{2} - 12 x - 12$

Step 5.4.2.4

Subtract $4 y$ from both sides of the equation.

$3 y^{2} y' {(x + 2)}^{2} - 4 x y' - 8 y' = 20 x^{3} + 80 x^{2} + 80 x - 3 x^{2} - 12 x - 12 - 4 y$

Step 5.4.2.5

Subtract $3 x^{2}$ from $80 x^{2}$ .

$3 y^{2} y' {(x + 2)}^{2} - 4 x y' - 8 y' = 20 x^{3} + 77 x^{2} + 80 x - 12 x - 12 - 4 y$

Step 5.4.2.6

Subtract $12 x$ from $80 x$ .

$3 y^{2} y' {(x + 2)}^{2} - 4 x y' - 8 y' = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.3

Factor $y'$ out of $3 y^{2} y' {(x + 2)}^{2} - 4 x y' - 8 y'$ .

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Step 5.4.3.1

Factor $y'$ out of $3 y^{2} y' {(x + 2)}^{2}$ .

$y' (3 y^{2} {(x + 2)}^{2}) - 4 x y' - 8 y' = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.3.2

Factor $y'$ out of $- 4 x y'$ .

$y' (3 y^{2} {(x + 2)}^{2}) + y' (- 4 x) - 8 y' = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.3.3

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

$y' (3 y^{2} {(x + 2)}^{2}) + y' (- 4 x) + y' \cdot - 8 = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.3.4

Factor $y'$ out of $y' (3 y^{2} {(x + 2)}^{2}) + y' (- 4 x)$ .

$y' (3 y^{2} {(x + 2)}^{2} - 4 x) + y' \cdot - 8 = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.3.5

Factor $y'$ out of $y' (3 y^{2} {(x + 2)}^{2} - 4 x) + y' \cdot - 8$ .

$y' (3 y^{2} {(x + 2)}^{2} - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.4

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$y' (3 y^{2} ((x + 2) (x + 2)) - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.5

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 5.4.5.1

Apply the distributive property.

$y' (3 y^{2} (x (x + 2) + 2 (x + 2)) - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.5.2

Apply the distributive property.

$y' (3 y^{2} (x \cdot x + x \cdot 2 + 2 (x + 2)) - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.5.3

Apply the distributive property.

$y' (3 y^{2} (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.6

Simplify and combine like terms.

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Step 5.4.6.1

Simplify each term.

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Step 5.4.6.1.1

Multiply $x$ by $x$ .

$y' (3 y^{2} (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.6.1.2

Move $2$ to the left of $x$ .

$y' (3 y^{2} (x^{2} + 2 x + 2 x + 2 \cdot 2) - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.6.1.3

Multiply $2$ by $2$ .

$y' (3 y^{2} (x^{2} + 2 x + 2 x + 4) - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.6.2

Add $2 x$ and $2 x$ .

$y' (3 y^{2} (x^{2} + 4 x + 4) - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.7

Apply the distributive property.

$y' (3 y^{2} x^{2} + 3 y^{2} (4 x) + 3 y^{2} \cdot 4 - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.8

Simplify.

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Step 5.4.8.1

Rewrite using the commutative property of multiplication.

$y' (3 y^{2} x^{2} + 3 \cdot (4 y^{2} x) + 3 y^{2} \cdot 4 - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.8.2

Multiply $4$ by $3$ .

$y' (3 y^{2} x^{2} + 3 \cdot (4 y^{2} x) + 12 y^{2} - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.9

Multiply $3$ by $4$ .

$y' (3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$

Step 5.4.10

Divide each term in $y' (3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$ by $3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8$ and simplify.

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Step 5.4.10.1

Divide each term in $y' (3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8) = 20 x^{3} + 77 x^{2} + 68 x - 12 - 4 y$ by $3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8$ .

$\frac{y' (3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} = \frac{20 x^{3}}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{77 x^{2}}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{68 x}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{- 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{- 4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.2

Simplify the left side.

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Step 5.4.10.2.1

Cancel the common factor of $3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8$ .

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Step 5.4.10.2.1.1

Cancel the common factor.

Step 5.4.10.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{20 x^{3}}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{77 x^{2}}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{68 x}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{- 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{- 4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3

Simplify the right side.

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Step 5.4.10.3.1

Simplify terms.

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Step 5.4.10.3.1.1

Simplify each term.

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Step 5.4.10.3.1.1.1

Move the negative in front of the fraction.

$y' = \frac{20 x^{3}}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{77 x^{2}}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{68 x}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{- 4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.1.1.2

Move the negative in front of the fraction.

$y' = \frac{20 x^{3}}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{77 x^{2}}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{68 x}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.1.2

Simplify terms.

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Step 5.4.10.3.1.2.1

Combine the numerators over the common denominator.

$y' = \frac{20 x^{3} + 77 x^{2}}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{68 x}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.1.2.2

Factor $x^{2}$ out of $20 x^{3} + 77 x^{2}$ .

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Step 5.4.10.3.1.2.2.1

Factor $x^{2}$ out of $20 x^{3}$ .

$y' = \frac{x^{2} (20 x) + 77 x^{2}}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{68 x}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.1.2.2.2

Factor $x^{2}$ out of $77 x^{2}$ .

$y' = \frac{x^{2} (20 x) + x^{2} \cdot 77}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{68 x}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.1.2.2.3

Factor $x^{2}$ out of $x^{2} (20 x) + x^{2} \cdot 77$ .

$y' = \frac{x^{2} (20 x + 77)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} + \frac{68 x}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.1.2.3

Combine the numerators over the common denominator.

$y' = \frac{x^{2} (20 x + 77) + 68 x}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.2

Simplify the numerator.

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Step 5.4.10.3.2.1

Factor $x$ out of $x^{2} (20 x + 77) + 68 x$ .

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Step 5.4.10.3.2.1.1

Factor $x$ out of $x^{2} (20 x + 77)$ .

$y' = \frac{x (x (20 x + 77)) + 68 x}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.2.1.2

Factor $x$ out of $68 x$ .

$y' = \frac{x (x (20 x + 77)) + x \cdot 68}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.2.1.3

Factor $x$ out of $x (x (20 x + 77)) + x \cdot 68$ .

$y' = \frac{x (x (20 x + 77) + 68)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.2.2

Apply the distributive property.

$y' = \frac{x (x (20 x) + x \cdot 77 + 68)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.2.3

Rewrite using the commutative property of multiplication.

$y' = \frac{x (20 x \cdot x + x \cdot 77 + 68)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.2.4

Move $77$ to the left of $x$ .

$y' = \frac{x (20 x \cdot x + 77 \cdot x + 68)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.2.5

Multiply $x$ by $x$ by adding the exponents.

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Step 5.4.10.3.2.5.1

Move $x$ .

$y' = \frac{x (20 (x \cdot x) + 77 \cdot x + 68)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.2.5.2

Multiply $x$ by $x$ .

$y' = \frac{x (20 x^{2} + 77 \cdot x + 68)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

$y' = \frac{x (20 x^{2} + 77 x + 68)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.3

Combine the numerators over the common denominator.

$y' = \frac{x (20 x^{2} + 77 x + 68) - 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4

Simplify the numerator.

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Step 5.4.10.3.4.1

Apply the distributive property.

$y' = \frac{x (20 x^{2}) + x (77 x) + x \cdot 68 - 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.2

Simplify.

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Step 5.4.10.3.4.2.1

Rewrite using the commutative property of multiplication.

$y' = \frac{20 x \cdot x^{2} + x (77 x) + x \cdot 68 - 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.2.2

Rewrite using the commutative property of multiplication.

$y' = \frac{20 x \cdot x^{2} + 77 x \cdot x + x \cdot 68 - 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.2.3

Move $68$ to the left of $x$ .

$y' = \frac{20 x \cdot x^{2} + 77 x \cdot x + 68 \cdot x - 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.3

Simplify each term.

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Step 5.4.10.3.4.3.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.4.10.3.4.3.1.1

Move $x^{2}$ .

$y' = \frac{20 (x^{2} x) + 77 x \cdot x + 68 \cdot x - 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.3.1.2

Multiply $x^{2}$ by $x$ .

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Step 5.4.10.3.4.3.1.2.1

Raise $x$ to the power of $1$ .

$y' = \frac{20 (x^{2} x^{1}) + 77 x \cdot x + 68 \cdot x - 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.3.1.2.2

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

$y' = \frac{20 x^{2 + 1} + 77 x \cdot x + 68 \cdot x - 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.3.1.3

Add $2$ and $1$ .

$y' = \frac{20 x^{3} + 77 x \cdot x + 68 \cdot x - 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.3.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.4.10.3.4.3.2.1

Move $x$ .

$y' = \frac{20 x^{3} + 77 (x \cdot x) + 68 \cdot x - 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.3.2.2

Multiply $x$ by $x$ .

$y' = \frac{20 x^{3} + 77 x^{2} + 68 \cdot x - 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

$y' = \frac{20 x^{3} + 77 x^{2} + 68 x - 12}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.4

Rewrite $20 x^{3} + 77 x^{2} + 68 x - 12$ in a factored form.

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Step 5.4.10.3.4.4.1

Factor $20 x^{3} + 77 x^{2} + 68 x - 12$ using the rational roots test.

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Step 5.4.10.3.4.4.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

$q = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

Step 5.4.10.3.4.4.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.05, \pm 0.5, \pm 0.1, \pm 0.25, \pm 0.2, \pm 12, \pm 0.6, \pm 6, \pm 1.2, \pm 3, \pm 2.4, \pm 2, \pm 0.4, \pm 0.3, \pm 1.5, \pm 0.15, \pm 0.75, \pm 4, \pm 0.8$

Step 5.4.10.3.4.4.1.3

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

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Step 5.4.10.3.4.4.1.3.1

Substitute $- 2$ into the polynomial.

$20 {(- 2)}^{3} + 77 {(- 2)}^{2} + 68 \cdot - 2 - 12$

Step 5.4.10.3.4.4.1.3.2

Raise $- 2$ to the power of $3$ .

$20 \cdot - 8 + 77 {(- 2)}^{2} + 68 \cdot - 2 - 12$

Step 5.4.10.3.4.4.1.3.3

Multiply $20$ by $- 8$ .

$- 160 + 77 {(- 2)}^{2} + 68 \cdot - 2 - 12$

Step 5.4.10.3.4.4.1.3.4

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

$- 160 + 77 \cdot 4 + 68 \cdot - 2 - 12$

Step 5.4.10.3.4.4.1.3.5

Multiply $77$ by $4$ .

$- 160 + 308 + 68 \cdot - 2 - 12$

Step 5.4.10.3.4.4.1.3.6

Add $- 160$ and $308$ .

$148 + 68 \cdot - 2 - 12$

Step 5.4.10.3.4.4.1.3.7

Multiply $68$ by $- 2$ .

$148 - 136 - 12$

Step 5.4.10.3.4.4.1.3.8

Subtract $136$ from $148$ .

$12 - 12$

Step 5.4.10.3.4.4.1.3.9

Subtract $12$ from $12$ .

$0$

Step 5.4.10.3.4.4.1.4

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{20 x^{3} + 77 x^{2} + 68 x - 12}{x + 2}$

Step 5.4.10.3.4.4.1.5

Divide $20 x^{3} + 77 x^{2} + 68 x - 12$ by $x + 2$ .

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Step 5.4.10.3.4.4.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$20 x^{3}$

$77 x^{2}$

$68 x$

$12$

Step 5.4.10.3.4.4.1.5.2

Divide the highest order term in the dividend $20 x^{3}$ by the highest order term in divisor $x$ .

					$20 x^{2}$
	$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$

Step 5.4.10.3.4.4.1.5.3

Multiply the new quotient term by the divisor.

				$20 x^{2}$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			+	$20 x^{3}$	+	$40 x^{2}$

Step 5.4.10.3.4.4.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $20 x^{3} + 40 x^{2}$

				$20 x^{2}$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			-	$20 x^{3}$	-	$40 x^{2}$

Step 5.4.10.3.4.4.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$20 x^{2}$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			-	$20 x^{3}$	-	$40 x^{2}$
					+	$37 x^{2}$

Step 5.4.10.3.4.4.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$20 x^{2}$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			-	$20 x^{3}$	-	$40 x^{2}$
					+	$37 x^{2}$	+	$68 x$

Step 5.4.10.3.4.4.1.5.7

Divide the highest order term in the dividend $37 x^{2}$ by the highest order term in divisor $x$ .

				$20 x^{2}$	+	$37 x$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			-	$20 x^{3}$	-	$40 x^{2}$
					+	$37 x^{2}$	+	$68 x$

Step 5.4.10.3.4.4.1.5.8

Multiply the new quotient term by the divisor.

				$20 x^{2}$	+	$37 x$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			-	$20 x^{3}$	-	$40 x^{2}$
					+	$37 x^{2}$	+	$68 x$
					+	$37 x^{2}$	+	$74 x$

Step 5.4.10.3.4.4.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $37 x^{2} + 74 x$

				$20 x^{2}$	+	$37 x$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			-	$20 x^{3}$	-	$40 x^{2}$
					+	$37 x^{2}$	+	$68 x$
					-	$37 x^{2}$	-	$74 x$

Step 5.4.10.3.4.4.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$20 x^{2}$	+	$37 x$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			-	$20 x^{3}$	-	$40 x^{2}$
					+	$37 x^{2}$	+	$68 x$
					-	$37 x^{2}$	-	$74 x$
							-	$6 x$

Step 5.4.10.3.4.4.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$20 x^{2}$	+	$37 x$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			-	$20 x^{3}$	-	$40 x^{2}$
					+	$37 x^{2}$	+	$68 x$
					-	$37 x^{2}$	-	$74 x$
							-	$6 x$	-	$12$

Step 5.4.10.3.4.4.1.5.12

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

				$20 x^{2}$	+	$37 x$	-	$6$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			-	$20 x^{3}$	-	$40 x^{2}$
					+	$37 x^{2}$	+	$68 x$
					-	$37 x^{2}$	-	$74 x$
							-	$6 x$	-	$12$

Step 5.4.10.3.4.4.1.5.13

Multiply the new quotient term by the divisor.

				$20 x^{2}$	+	$37 x$	-	$6$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			-	$20 x^{3}$	-	$40 x^{2}$
					+	$37 x^{2}$	+	$68 x$
					-	$37 x^{2}$	-	$74 x$
							-	$6 x$	-	$12$
							-	$6 x$	-	$12$

Step 5.4.10.3.4.4.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x - 12$

				$20 x^{2}$	+	$37 x$	-	$6$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			-	$20 x^{3}$	-	$40 x^{2}$
					+	$37 x^{2}$	+	$68 x$
					-	$37 x^{2}$	-	$74 x$
							-	$6 x$	-	$12$
							+	$6 x$	+	$12$

Step 5.4.10.3.4.4.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$20 x^{2}$	+	$37 x$	-	$6$
$x$	+	$2$		$20 x^{3}$	+	$77 x^{2}$	+	$68 x$	-	$12$
			-	$20 x^{3}$	-	$40 x^{2}$
					+	$37 x^{2}$	+	$68 x$
					-	$37 x^{2}$	-	$74 x$
							-	$6 x$	-	$12$
							+	$6 x$	+	$12$
										$0$

Step 5.4.10.3.4.4.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$20 x^{2} + 37 x - 6$

Step 5.4.10.3.4.4.1.6

Write $20 x^{3} + 77 x^{2} + 68 x - 12$ as a set of factors.

$y' = \frac{(x + 2) (20 x^{2} + 37 x - 6)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.4.2

Factor by grouping.

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Step 5.4.10.3.4.4.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 20 \cdot - 6 = - 120$ and whose sum is $b = 37$ .

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Step 5.4.10.3.4.4.2.1.1

Factor $37$ out of $37 x$ .

$y' = \frac{(x + 2) (20 x^{2} + 37 (x) - 6)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.4.2.1.2

Rewrite $37$ as $- 3$ plus $40$

$y' = \frac{(x + 2) (20 x^{2} + (- 3 + 40) x - 6)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.4.2.1.3

Apply the distributive property.

$y' = \frac{(x + 2) (20 x^{2} - 3 x + 40 x - 6)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.4.2.2

Factor out the greatest common factor from each group.

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Step 5.4.10.3.4.4.2.2.1

Group the first two terms and the last two terms.

$y' = \frac{(x + 2) ((20 x^{2} - 3 x) + 40 x - 6)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.4.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y' = \frac{(x + 2) (x (20 x - 3) + 2 (20 x - 3))}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.4.2.3

Factor the polynomial by factoring out the greatest common factor, $20 x - 3$ .

$y' = \frac{(x + 2) ((20 x - 3) (x + 2))}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

$y' = \frac{(x + 2) (20 x - 3) (x + 2)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.5

Combine exponents.

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Step 5.4.10.3.4.5.1

Raise $x + 2$ to the power of $1$ .

$y' = \frac{{(x + 2)}^{1} (x + 2) (20 x - 3)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.5.2

Raise $x + 2$ to the power of $1$ .

$y' = \frac{{(x + 2)}^{1} {(x + 2)}^{1} (20 x - 3)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.5.3

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

$y' = \frac{{(x + 2)}^{1 + 1} (20 x - 3)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.4.5.4

Add $1$ and $1$ .

$y' = \frac{{(x + 2)}^{2} (20 x - 3)}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8} - \frac{4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.5

Combine the numerators over the common denominator.

$y' = \frac{{(x + 2)}^{2} (20 x - 3) - 4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.6

Simplify the numerator.

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Step 5.4.10.3.6.1

Apply the distributive property.

$y' = \frac{{(x + 2)}^{2} (20 x) + {(x + 2)}^{2} \cdot - 3 - 4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.6.2

Rewrite using the commutative property of multiplication.

$y' = \frac{20 {(x + 2)}^{2} x + {(x + 2)}^{2} \cdot - 3 - 4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.6.3

Move $- 3$ to the left of ${(x + 2)}^{2}$ .

$y' = \frac{20 {(x + 2)}^{2} x - 3 {(x + 2)}^{2} - 4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 5.4.10.3.7

Reorder factors in $\frac{20 {(x + 2)}^{2} x - 3 {(x + 2)}^{2} - 4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$ .

$y' = \frac{20 x {(x + 2)}^{2} - 3 {(x + 2)}^{2} - 4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$

Step 6

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

$\frac{d y}{d x} = \frac{20 x {(x + 2)}^{2} - 3 {(x + 2)}^{2} - 4 y}{3 y^{2} x^{2} + 12 y^{2} x + 12 y^{2} - 4 x - 8}$