Algebra Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

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

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

Step 2.3

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

$2 x x' + 2 y$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $2 x^{2} + 2 y^{2} - x$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u_{1}$ with $2 x^{2} + 2 y^{2} - x$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

Multiply $2$ by $2$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Multiply $2$ by $2$ .

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

Step 3.9

Since $- 1$ is constant with respect to $y$ , the derivative of $- x$ with respect to $y$ is $- \frac{d}{d y} [x]$ .

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

Step 3.10

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

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

Step 3.11

Simplify.

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Step 3.11.1

Apply the distributive property.

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

Step 3.11.2

Combine terms.

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Step 3.11.2.1

Multiply $2$ by $2$ .

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

Step 3.11.2.2

Multiply $2$ by $2$ .

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

Step 3.11.2.3

Multiply $- 1$ by $2$ .

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

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Since $x'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 5.2

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

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Step 5.2.1

Rewrite.

$0 + 0 + (4 x^{2} + 4 y^{2} - 2 x) (4 x x' + 4 y - x') = 2 x x' + 2 y$

Step 5.2.2

Simplify by adding zeros.

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

Step 5.2.3

Expand $(4 x^{2} + 4 y^{2} - 2 x) (4 x x' + 4 y - x')$ by multiplying each term in the first expression by each term in the second expression.

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

Step 5.2.4

Simplify terms.

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Step 5.2.4.1

Simplify each term.

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Step 5.2.4.1.1

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

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Step 5.2.4.1.1.1

Move $x$ .

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

Step 5.2.4.1.1.2

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

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Step 5.2.4.1.1.2.1

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

$4 (x^{1} x^{2}) (4 x') + 4 x^{2} (4 y) + 4 x^{2} (- x') + 4 y^{2} (4 x x') + 4 y^{2} (4 y) + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.1.2.2

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

$4 x^{1 + 2} (4 x') + 4 x^{2} (4 y) + 4 x^{2} (- x') + 4 y^{2} (4 x x') + 4 y^{2} (4 y) + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.1.3

Add $1$ and $2$ .

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

Step 5.2.4.1.2

Rewrite using the commutative property of multiplication.

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

Step 5.2.4.1.3

Multiply $4$ by $4$ .

$16 x^{3} x' + 4 x^{2} (4 y) + 4 x^{2} (- x') + 4 y^{2} (4 x x') + 4 y^{2} (4 y) + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.4

Rewrite using the commutative property of multiplication.

$16 x^{3} x' + 4 \cdot 4 x^{2} y + 4 x^{2} (- x') + 4 y^{2} (4 x x') + 4 y^{2} (4 y) + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.5

Multiply $4$ by $4$ .

$16 x^{3} x' + 16 x^{2} y + 4 x^{2} (- x') + 4 y^{2} (4 x x') + 4 y^{2} (4 y) + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.6

Rewrite using the commutative property of multiplication.

$16 x^{3} x' + 16 x^{2} y + 4 \cdot - 1 x^{2} x' + 4 y^{2} (4 x x') + 4 y^{2} (4 y) + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.7

Multiply $4$ by $- 1$ .

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 4 y^{2} (4 x x') + 4 y^{2} (4 y) + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.8

Multiply $4$ by $4$ .

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} (x x') + 4 y^{2} (4 y) + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.9

Rewrite using the commutative property of multiplication.

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 4 \cdot 4 y^{2} y + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.10

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

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Step 5.2.4.1.10.1

Move $y$ .

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 4 \cdot 4 (y \cdot y^{2}) + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.10.2

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

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Step 5.2.4.1.10.2.1

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

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 4 \cdot 4 (y^{1} y^{2}) + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.10.2.2

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

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 4 \cdot 4 y^{1 + 2} + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.10.3

Add $1$ and $2$ .

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 4 \cdot 4 y^{3} + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.11

Multiply $4$ by $4$ .

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 16 y^{3} + 4 y^{2} (- x') - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.12

Rewrite using the commutative property of multiplication.

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 16 y^{3} + 4 \cdot - 1 y^{2} x' - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.13

Multiply $4$ by $- 1$ .

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 2 x (4 x x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.14

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

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Step 5.2.4.1.14.1

Move $x$ .

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 2 (x \cdot x) (4 x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.14.2

Multiply $x$ by $x$ .

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 2 x^{2} (4 x') - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.15

Rewrite using the commutative property of multiplication.

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 2 \cdot 4 x^{2} x' - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.16

Multiply $- 2$ by $4$ .

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 8 x^{2} x' - 2 x (4 y) - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.17

Rewrite using the commutative property of multiplication.

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 8 x^{2} x' - 2 \cdot 4 x y - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.18

Multiply $- 2$ by $4$ .

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 8 x^{2} x' - 8 x y - 2 x (- x') = 2 x x' + 2 y$

Step 5.2.4.1.19

Rewrite using the commutative property of multiplication.

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 8 x^{2} x' - 8 x y - 2 \cdot - 1 x x' = 2 x x' + 2 y$

Step 5.2.4.1.20

Multiply $- 2$ by $- 1$ .

$16 x^{3} x' + 16 x^{2} y - 4 x^{2} x' + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 8 x^{2} x' - 8 x y + 2 x x' = 2 x x' + 2 y$

Step 5.2.4.2

Subtract $8 x^{2} x'$ from $- 4 x^{2} x'$ .

$16 x^{3} x' + 16 x^{2} y + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 12 x^{2} x' - 8 x y + 2 x x' = 2 x x' + 2 y$

Step 5.3

Move all terms containing $x'$ to the left side of the equation.

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Step 5.3.1

Subtract $2 x x'$ from both sides of the equation.

$16 x^{3} x' + 16 x^{2} y + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 12 x^{2} x' - 8 x y + 2 x x' - 2 x x' = 2 y$

Step 5.3.2

Combine the opposite terms in $16 x^{3} x' + 16 x^{2} y + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 12 x^{2} x' - 8 x y + 2 x x' - 2 x x'$ .

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Step 5.3.2.1

Subtract $2 x x'$ from $2 x x'$ .

$16 x^{3} x' + 16 x^{2} y + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 12 x^{2} x' - 8 x y + 0 = 2 y$

Step 5.3.2.2

Add $16 x^{3} x' + 16 x^{2} y + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 12 x^{2} x' - 8 x y$ and $0$ .

$16 x^{3} x' + 16 x^{2} y + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 12 x^{2} x' - 8 x y = 2 y$

Step 5.4

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

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Step 5.4.1

Subtract $16 x^{2} y$ from both sides of the equation.

$16 x^{3} x' + 16 y^{2} x x' + 16 y^{3} - 4 y^{2} x' - 12 x^{2} x' - 8 x y = 2 y - 16 x^{2} y$

Step 5.4.2

Subtract $16 y^{3}$ from both sides of the equation.

$16 x^{3} x' + 16 y^{2} x x' - 4 y^{2} x' - 12 x^{2} x' - 8 x y = 2 y - 16 x^{2} y - 16 y^{3}$

Step 5.4.3

Add $8 x y$ to both sides of the equation.

$16 x^{3} x' + 16 y^{2} x x' - 4 y^{2} x' - 12 x^{2} x' = 2 y - 16 x^{2} y - 16 y^{3} + 8 x y$

Step 5.5

Factor $4 x'$ out of $16 x^{3} x' + 16 y^{2} x x' - 4 y^{2} x' - 12 x^{2} x'$ .

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Step 5.5.1

Factor $4 x'$ out of $16 x^{3} x'$ .

$4 x' (4 x^{3}) + 16 y^{2} x x' - 4 y^{2} x' - 12 x^{2} x' = 2 y - 16 x^{2} y - 16 y^{3} + 8 x y$

Step 5.5.2

Factor $4 x'$ out of $16 y^{2} x x'$ .

$4 x' (4 x^{3}) + 4 x' (4 y^{2} x) - 4 y^{2} x' - 12 x^{2} x' = 2 y - 16 x^{2} y - 16 y^{3} + 8 x y$

Step 5.5.3

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

$4 x' (4 x^{3}) + 4 x' (4 y^{2} x) + 4 x' (- y^{2}) - 12 x^{2} x' = 2 y - 16 x^{2} y - 16 y^{3} + 8 x y$

Step 5.5.4

Factor $4 x'$ out of $- 12 x^{2} x'$ .

$4 x' (4 x^{3}) + 4 x' (4 y^{2} x) + 4 x' (- y^{2}) + 4 x' (- 3 x^{2}) = 2 y - 16 x^{2} y - 16 y^{3} + 8 x y$

Step 5.5.5

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

$4 x' (4 x^{3} + 4 y^{2} x) + 4 x' (- y^{2}) + 4 x' (- 3 x^{2}) = 2 y - 16 x^{2} y - 16 y^{3} + 8 x y$

Step 5.5.6

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

$4 x' (4 x^{3} + 4 y^{2} x - y^{2}) + 4 x' (- 3 x^{2}) = 2 y - 16 x^{2} y - 16 y^{3} + 8 x y$

Step 5.5.7

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

$4 x' (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}) = 2 y - 16 x^{2} y - 16 y^{3} + 8 x y$

Step 5.6

Divide each term in $4 x' (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}) = 2 y - 16 x^{2} y - 16 y^{3} + 8 x y$ by $4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})$ and simplify.

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Step 5.6.1

Divide each term in $4 x' (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}) = 2 y - 16 x^{2} y - 16 y^{3} + 8 x y$ by $4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})$ .

$\frac{4 x' (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} = \frac{2 y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 x^{2} y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 y^{3}}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{8 x y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})}$

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $4$ .

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Step 5.6.2.1.1

Cancel the common factor.

Step 5.6.2.1.2

Rewrite the expression.

$\frac{x' (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})}{4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}} = \frac{2 y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 x^{2} y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 y^{3}}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{8 x y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})}$

Step 5.6.2.2

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

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Step 5.6.2.2.1

Cancel the common factor.

Step 5.6.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{2 y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 x^{2} y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 y^{3}}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{8 x y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})}$

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Simplify each term.

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Step 5.6.3.1.1

Cancel the common factor of $2$ and $4$ .

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Step 5.6.3.1.1.1

Factor $2$ out of $2 y$ .

$x' = \frac{2 (y)}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 x^{2} y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 y^{3}}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{8 x y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})}$

Step 5.6.3.1.1.2

Cancel the common factors.

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Step 5.6.3.1.1.2.1

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

$x' = \frac{2 (y)}{2 (2 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}))} + \frac{- 16 x^{2} y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 y^{3}}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{8 x y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})}$

Step 5.6.3.1.1.2.2

Cancel the common factor.

$x' = \frac{2 y}{2 (2 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}))} + \frac{- 16 x^{2} y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 y^{3}}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{8 x y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})}$

Step 5.6.3.1.1.2.3

Rewrite the expression.

$x' = \frac{y}{2 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 x^{2} y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 y^{3}}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{8 x y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})}$

Step 5.6.3.1.2

Cancel the common factor of $- 16$ and $4$ .

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Step 5.6.3.1.2.1

Factor $4$ out of $- 16 x^{2} y$ .

$x' = \frac{y}{2 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{4 (- 4 x^{2} y)}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 16 y^{3}}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{8 x y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})}$

Step 5.6.3.1.2.2

Cancel the common factors.

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Step 5.6.3.1.2.2.1

Cancel the common factor.

Step 5.6.3.1.2.2.2

Rewrite the expression.

$x' = \frac{y}{2 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{- 4 x^{2} y}{4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}} + \frac{- 16 y^{3}}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{8 x y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})}$

Step 5.6.3.1.3

Move the negative in front of the fraction.

$x' = \frac{y}{2 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} - \frac{4 x^{2} y}{4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}} + \frac{- 16 y^{3}}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})} + \frac{8 x y}{4 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})}$

Step 5.6.3.1.4

Cancel the common factor of $- 16$ and $4$ .

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Step 5.6.3.1.4.1

Factor $4$ out of $- 16 y^{3}$ .

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

Step 5.6.3.1.4.2

Cancel the common factors.

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Step 5.6.3.1.4.2.1

Cancel the common factor.

Step 5.6.3.1.4.2.2

Rewrite the expression.

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

Step 5.6.3.1.5

Move the negative in front of the fraction.

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

Step 5.6.3.1.6

Cancel the common factor of $8$ and $4$ .

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Step 5.6.3.1.6.1

Factor $4$ out of $8 x y$ .

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

Step 5.6.3.1.6.2

Cancel the common factors.

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Step 5.6.3.1.6.2.1

Cancel the common factor.

Step 5.6.3.1.6.2.2

Rewrite the expression.

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

Step 5.6.3.2

To write $- \frac{4 x^{2} y}{4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.6.3.3

Write each expression with a common denominator of $2 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.6.3.3.1

Multiply $\frac{4 x^{2} y}{4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}}$ by $\frac{2}{2}$ .

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

Step 5.6.3.3.2

Reorder the factors of $(4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}) \cdot 2$ .

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

Step 5.6.3.4

Combine the numerators over the common denominator.

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

Step 5.6.3.5

Simplify the numerator.

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Step 5.6.3.5.1

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

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Step 5.6.3.5.1.1

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

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

Step 5.6.3.5.1.2

Factor $y$ out of $y^{1}$ .

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

Step 5.6.3.5.1.3

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

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

Step 5.6.3.5.1.4

Factor $y$ out of $y \cdot 1 + y (- 4 x^{2} \cdot 2)$ .

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

Step 5.6.3.5.2

Multiply $2$ by $- 4$ .

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

Step 5.6.3.6

To write $- \frac{4 y^{3}}{4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.6.3.7

Write each expression with a common denominator of $2 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.6.3.7.1

Multiply $\frac{4 y^{3}}{4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}}$ by $\frac{2}{2}$ .

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

Step 5.6.3.7.2

Reorder the factors of $(4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}) \cdot 2$ .

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

Step 5.6.3.8

Combine the numerators over the common denominator.

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

Step 5.6.3.9

Simplify the numerator.

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Step 5.6.3.9.1

Factor $y$ out of $y (1 - 8 x^{2}) - 4 y^{3} \cdot 2$ .

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Step 5.6.3.9.1.1

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

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

Step 5.6.3.9.1.2

Factor $y$ out of $y (1 - 8 x^{2}) + y (- 4 y^{2} \cdot 2)$ .

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

Step 5.6.3.9.2

Multiply $2$ by $- 4$ .

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

Step 5.6.3.10

To write $\frac{2 x y}{4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.6.3.11

Write each expression with a common denominator of $2 (4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2})$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.6.3.11.1

Multiply $\frac{2 x y}{4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}}$ by $\frac{2}{2}$ .

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

Step 5.6.3.11.2

Reorder the factors of $(4 x^{3} + 4 y^{2} x - y^{2} - 3 x^{2}) \cdot 2$ .

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

Step 5.6.3.12

Combine the numerators over the common denominator.

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

Step 5.6.3.13

Simplify the numerator.

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Step 5.6.3.13.1

Factor $y$ out of $y (1 - 8 x^{2} - 8 y^{2}) + 2 x y \cdot 2$ .

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Step 5.6.3.13.1.1

Factor $y$ out of $2 x y \cdot 2$ .

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

Step 5.6.3.13.1.2

Factor $y$ out of $y (1 - 8 x^{2} - 8 y^{2}) + y (2 x \cdot 2)$ .

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

Step 5.6.3.13.2

Multiply $2$ by $2$ .

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

Step 6

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

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