Algebra Examples

${(x^{2} + y^{2})}^{3} = 8 x^{2} y^{2}$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

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

$\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [x^{2} + y^{2}]$

Step 2.1.2

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

$3 {u_{1}}^{2} \frac{d}{d x} [x^{2} + y^{2}]$

Step 2.1.3

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

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2} y^{2}$ with respect to $x$ is $8 \frac{d}{d x} [x^{2} y^{2}]$ .

$8 \frac{d}{d x} [x^{2} y^{2}]$

Step 3.2

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

$8 (x^{2} \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x^{2}])$

Step 3.3

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

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Step 3.3.1

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

$8 (x^{2} (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}])$

Step 3.3.2

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

$8 (x^{2} (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}])$

Step 3.3.3

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

$8 (x^{2} (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}])$

Step 3.4

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

$8 (2 \cdot x^{2} y \frac{d}{d x} [y] + y^{2} \frac{d}{d x} [x^{2}])$

Step 3.5

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

$8 (2 x^{2} y y' + y^{2} \frac{d}{d x} [x^{2}])$

Step 3.6

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

$8 (2 x^{2} y y' + y^{2} (2 x))$

Step 3.7

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

$8 (2 x^{2} y y' + 2 \cdot y^{2} x)$

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

$8 (2 x^{2} y y') + 8 (2 y^{2} x)$

Step 3.8.2

Combine terms.

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Step 3.8.2.1

Multiply $2$ by $8$ .

$16 (x^{2} y y') + 8 (2 y^{2} x)$

Step 3.8.2.2

Multiply $2$ by $8$ .

$16 x^{2} y y' + 16 y^{2} x$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Simplify $3 {(x^{2} + y^{2})}^{2} (2 x + 2 y y')$ .

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Step 5.1.1

Rewrite.

$0 + 0 + 3 {(x^{2} + y^{2})}^{2} (2 x + 2 y y') = 16 x^{2} y y' + 16 y^{2} x$

Step 5.1.2

Simplify by adding zeros.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.1.4

Simplify the expression.

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Step 5.1.4.1

Rewrite using the commutative property of multiplication.

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

Step 5.1.4.2

Multiply $2$ by $3$ .

$3 \cdot 2 {(x^{2} + y^{2})}^{2} x + 6 {(x^{2} + y^{2})}^{2} (y y') = 16 x^{2} y y' + 16 y^{2} x$

Step 5.1.4.3

Multiply $3$ by $2$ .

$6 {(x^{2} + y^{2})}^{2} x + 6 {(x^{2} + y^{2})}^{2} (y y') = 16 x^{2} y y' + 16 y^{2} x$

Step 5.1.4.4

Reorder factors in $6 {(x^{2} + y^{2})}^{2} x + 6 {(x^{2} + y^{2})}^{2} (y y')$ .

$6 x {(x^{2} + y^{2})}^{2} + 6 y y' {(x^{2} + y^{2})}^{2} = 16 x^{2} y y' + 16 y^{2} x$

Step 5.2

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

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Step 5.2.1

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

$6 x {(x^{2} + y^{2})}^{2} + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

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

$6 x ((x^{2} + y^{2}) (x^{2} + y^{2})) + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.2

Expand $(x^{2} + y^{2}) (x^{2} + y^{2})$ using the FOIL Method.

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Step 5.2.2.2.1

Apply the distributive property.

$6 x (x^{2} (x^{2} + y^{2}) + y^{2} (x^{2} + y^{2})) + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.2.2

Apply the distributive property.

$6 x (x^{2} x^{2} + x^{2} y^{2} + y^{2} (x^{2} + y^{2})) + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.2.3

Apply the distributive property.

$6 x (x^{2} x^{2} + x^{2} y^{2} + y^{2} x^{2} + y^{2} y^{2}) + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.3

Simplify and combine like terms.

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Step 5.2.2.3.1

Simplify each term.

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Step 5.2.2.3.1.1

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

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Step 5.2.2.3.1.1.1

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

$6 x (x^{2 + 2} + x^{2} y^{2} + y^{2} x^{2} + y^{2} y^{2}) + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.3.1.1.2

Add $2$ and $2$ .

$6 x (x^{4} + x^{2} y^{2} + y^{2} x^{2} + y^{2} y^{2}) + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.3.1.2

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

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Step 5.2.2.3.1.2.1

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

$6 x (x^{4} + x^{2} y^{2} + y^{2} x^{2} + y^{2 + 2}) + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.3.1.2.2

Add $2$ and $2$ .

$6 x (x^{4} + x^{2} y^{2} + y^{2} x^{2} + y^{4}) + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.3.2

Add $x^{2} y^{2}$ and $y^{2} x^{2}$ .

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Step 5.2.2.3.2.1

Reorder $y^{2}$ and $x^{2}$ .

$6 x (x^{4} + x^{2} y^{2} + x^{2} y^{2} + y^{4}) + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.3.2.2

Add $x^{2} y^{2}$ and $x^{2} y^{2}$ .

$6 x (x^{4} + 2 x^{2} y^{2} + y^{4}) + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.4

Apply the distributive property.

$6 x \cdot x^{4} + 6 x (2 x^{2} y^{2}) + 6 x y^{4} + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.5

Simplify.

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Step 5.2.2.5.1

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 5.2.2.5.1.1

Move $x^{4}$ .

$6 (x^{4} x) + 6 x (2 x^{2} y^{2}) + 6 x y^{4} + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.5.1.2

Multiply $x^{4}$ by $x$ .

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Step 5.2.2.5.1.2.1

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

$6 (x^{4} x^{1}) + 6 x (2 x^{2} y^{2}) + 6 x y^{4} + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.5.1.2.2

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

$6 x^{4 + 1} + 6 x (2 x^{2} y^{2}) + 6 x y^{4} + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.5.1.3

Add $4$ and $1$ .

$6 x^{5} + 6 x (2 x^{2} y^{2}) + 6 x y^{4} + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.5.2

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

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Step 5.2.2.5.2.1

Move $x^{2}$ .

$6 x^{5} + 6 (x^{2} x) (2 y^{2}) + 6 x y^{4} + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.5.2.2

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

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Step 5.2.2.5.2.2.1

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

$6 x^{5} + 6 (x^{2} x^{1}) (2 y^{2}) + 6 x y^{4} + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.5.2.2.2

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

$6 x^{5} + 6 x^{2 + 1} (2 y^{2}) + 6 x y^{4} + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.5.2.3

Add $2$ and $1$ .

$6 x^{5} + 6 x^{3} (2 y^{2}) + 6 x y^{4} + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.6

Simplify each term.

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Step 5.2.2.6.1

Rewrite using the commutative property of multiplication.

$6 x^{5} + 6 \cdot 2 x^{3} y^{2} + 6 x y^{4} + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.6.2

Multiply $6$ by $2$ .

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' {(x^{2} + y^{2})}^{2} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.7

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

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' ((x^{2} + y^{2}) (x^{2} + y^{2})) - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.8

Expand $(x^{2} + y^{2}) (x^{2} + y^{2})$ using the FOIL Method.

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Step 5.2.2.8.1

Apply the distributive property.

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' (x^{2} (x^{2} + y^{2}) + y^{2} (x^{2} + y^{2})) - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.8.2

Apply the distributive property.

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' (x^{2} x^{2} + x^{2} y^{2} + y^{2} (x^{2} + y^{2})) - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.8.3

Apply the distributive property.

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' (x^{2} x^{2} + x^{2} y^{2} + y^{2} x^{2} + y^{2} y^{2}) - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.9

Simplify and combine like terms.

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Step 5.2.2.9.1

Simplify each term.

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Step 5.2.2.9.1.1

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

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Step 5.2.2.9.1.1.1

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

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' (x^{2 + 2} + x^{2} y^{2} + y^{2} x^{2} + y^{2} y^{2}) - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.9.1.1.2

Add $2$ and $2$ .

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' (x^{4} + x^{2} y^{2} + y^{2} x^{2} + y^{2} y^{2}) - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.9.1.2

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

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Step 5.2.2.9.1.2.1

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

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' (x^{4} + x^{2} y^{2} + y^{2} x^{2} + y^{2 + 2}) - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.9.1.2.2

Add $2$ and $2$ .

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' (x^{4} + x^{2} y^{2} + y^{2} x^{2} + y^{4}) - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.9.2

Add $x^{2} y^{2}$ and $y^{2} x^{2}$ .

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Step 5.2.2.9.2.1

Reorder $y^{2}$ and $x^{2}$ .

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' (x^{4} + x^{2} y^{2} + x^{2} y^{2} + y^{4}) - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.9.2.2

Add $x^{2} y^{2}$ and $x^{2} y^{2}$ .

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' (x^{4} + 2 x^{2} y^{2} + y^{4}) - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.10

Apply the distributive property.

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' x^{4} + 6 y y' (2 x^{2} y^{2}) + 6 y y' y^{4} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.11

Simplify.

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Step 5.2.2.11.1

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

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Step 5.2.2.11.1.1

Move $y^{2}$ .

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' x^{4} + 6 (y^{2} y) y' (2 x^{2}) + 6 y y' y^{4} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.11.1.2

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

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Step 5.2.2.11.1.2.1

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

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' x^{4} + 6 (y^{2} y^{1}) y' (2 x^{2}) + 6 y y' y^{4} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.11.1.2.2

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

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' x^{4} + 6 y^{2 + 1} y' (2 x^{2}) + 6 y y' y^{4} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.11.1.3

Add $2$ and $1$ .

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' x^{4} + 6 y^{3} y' (2 x^{2}) + 6 y y' y^{4} - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.11.2

Multiply $y$ by $y^{4}$ by adding the exponents.

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Step 5.2.2.11.2.1

Move $y^{4}$ .

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' x^{4} + 6 y^{3} y' (2 x^{2}) + 6 (y^{4} y) y' - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.11.2.2

Multiply $y^{4}$ by $y$ .

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Step 5.2.2.11.2.2.1

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

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' x^{4} + 6 y^{3} y' (2 x^{2}) + 6 (y^{4} y^{1}) y' - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.11.2.2.2

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

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' x^{4} + 6 y^{3} y' (2 x^{2}) + 6 y^{4 + 1} y' - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.11.2.3

Add $4$ and $1$ .

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' x^{4} + 6 y^{3} y' (2 x^{2}) + 6 y^{5} y' - 16 x^{2} y y' = 16 y^{2} x$

Step 5.2.2.12

Multiply $2$ by $6$ .

$6 x^{5} + 12 x^{3} y^{2} + 6 x y^{4} + 6 y y' x^{4} + 12 y^{3} y' x^{2} + 6 y^{5} y' - 16 x^{2} y y' = 16 y^{2} x$

Step 5.3

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

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Step 5.3.1

Subtract $6 x^{5}$ from both sides of the equation.

$12 x^{3} y^{2} + 6 x y^{4} + 6 y y' x^{4} + 12 y^{3} y' x^{2} + 6 y^{5} y' - 16 x^{2} y y' = 16 y^{2} x - 6 x^{5}$

Step 5.3.2

Subtract $12 x^{3} y^{2}$ from both sides of the equation.

$6 x y^{4} + 6 y y' x^{4} + 12 y^{3} y' x^{2} + 6 y^{5} y' - 16 x^{2} y y' = 16 y^{2} x - 6 x^{5} - 12 x^{3} y^{2}$

Step 5.3.3

Subtract $6 x y^{4}$ from both sides of the equation.

$6 y y' x^{4} + 12 y^{3} y' x^{2} + 6 y^{5} y' - 16 x^{2} y y' = 16 y^{2} x - 6 x^{5} - 12 x^{3} y^{2} - 6 x y^{4}$

Step 5.4

Factor $2 y y'$ out of $6 y y' x^{4} + 12 y^{3} y' x^{2} + 6 y^{5} y' - 16 x^{2} y y'$ .

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Step 5.4.1

Factor $2 y y'$ out of $6 y y' x^{4}$ .

$2 y y' (3 x^{4}) + 12 y^{3} y' x^{2} + 6 y^{5} y' - 16 x^{2} y y' = 16 y^{2} x - 6 x^{5} - 12 x^{3} y^{2} - 6 x y^{4}$

Step 5.4.2

Factor $2 y y'$ out of $12 y^{3} y' x^{2}$ .

$2 y y' (3 x^{4}) + 2 y y' (6 y^{2} x^{2}) + 6 y^{5} y' - 16 x^{2} y y' = 16 y^{2} x - 6 x^{5} - 12 x^{3} y^{2} - 6 x y^{4}$

Step 5.4.3

Factor $2 y y'$ out of $6 y^{5} y'$ .

$2 y y' (3 x^{4}) + 2 y y' (6 y^{2} x^{2}) + 2 y y' (3 y^{4}) - 16 x^{2} y y' = 16 y^{2} x - 6 x^{5} - 12 x^{3} y^{2} - 6 x y^{4}$

Step 5.4.4

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

$2 y y' (3 x^{4}) + 2 y y' (6 y^{2} x^{2}) + 2 y y' (3 y^{4}) + 2 y y' (- 8 x^{2}) = 16 y^{2} x - 6 x^{5} - 12 x^{3} y^{2} - 6 x y^{4}$

Step 5.4.5

Factor $2 y y'$ out of $2 y y' (3 x^{4}) + 2 y y' (6 y^{2} x^{2})$ .

$2 y y' (3 x^{4} + 6 y^{2} x^{2}) + 2 y y' (3 y^{4}) + 2 y y' (- 8 x^{2}) = 16 y^{2} x - 6 x^{5} - 12 x^{3} y^{2} - 6 x y^{4}$

Step 5.4.6

Factor $2 y y'$ out of $2 y y' (3 x^{4} + 6 y^{2} x^{2}) + 2 y y' (3 y^{4})$ .

$2 y y' (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4}) + 2 y y' (- 8 x^{2}) = 16 y^{2} x - 6 x^{5} - 12 x^{3} y^{2} - 6 x y^{4}$

Step 5.4.7

Factor $2 y y'$ out of $2 y y' (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4}) + 2 y y' (- 8 x^{2})$ .

$2 y y' (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}) = 16 y^{2} x - 6 x^{5} - 12 x^{3} y^{2} - 6 x y^{4}$

Step 5.5

Divide each term in $2 y y' (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}) = 16 y^{2} x - 6 x^{5} - 12 x^{3} y^{2} - 6 x y^{4}$ by $2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})$ and simplify.

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Step 5.5.1

Divide each term in $2 y y' (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}) = 16 y^{2} x - 6 x^{5} - 12 x^{3} y^{2} - 6 x y^{4}$ by $2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})$ .

$\frac{2 y y' (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} = \frac{16 y^{2} x}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x^{5}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $2$ .

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Step 5.5.2.1.1

Cancel the common factor.

Step 5.5.2.1.2

Rewrite the expression.

$\frac{y y' (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} = \frac{16 y^{2} x}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x^{5}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.2.2

Cancel the common factor of $y$ .

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Step 5.5.2.2.1

Cancel the common factor.

Step 5.5.2.2.2

Rewrite the expression.

$\frac{y' (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} = \frac{16 y^{2} x}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x^{5}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.2.3

Cancel the common factor of $3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}$ .

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Step 5.5.2.3.1

Cancel the common factor.

Step 5.5.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{16 y^{2} x}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x^{5}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Cancel the common factor of $16$ and $2$ .

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Step 5.5.3.1.1.1

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

$y' = \frac{2 (8 y^{2} x)}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x^{5}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.1.2

Cancel the common factors.

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Step 5.5.3.1.1.2.1

Factor $2$ out of $2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})$ .

$y' = \frac{2 (8 y^{2} x)}{2 (y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}))} + \frac{- 6 x^{5}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.1.2.2

Cancel the common factor.

Step 5.5.3.1.1.2.3

Rewrite the expression.

$y' = \frac{8 y^{2} x}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x^{5}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.2

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

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Step 5.5.3.1.2.1

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

$y' = \frac{y (8 y x)}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x^{5}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.2.2

Cancel the common factors.

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Step 5.5.3.1.2.2.1

Cancel the common factor.

Step 5.5.3.1.2.2.2

Rewrite the expression.

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{- 6 x^{5}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.3

Cancel the common factor of $- 6$ and $2$ .

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Step 5.5.3.1.3.1

Factor $2$ out of $- 6 x^{5}$ .

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{2 (- 3 x^{5})}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.3.2

Cancel the common factors.

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Step 5.5.3.1.3.2.1

Factor $2$ out of $2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})$ .

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{2 (- 3 x^{5})}{2 (y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}))} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.3.2.2

Cancel the common factor.

Step 5.5.3.1.3.2.3

Rewrite the expression.

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{- 3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.4

Move the negative in front of the fraction.

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 12 x^{3} y^{2}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.5

Cancel the common factor of $- 12$ and $2$ .

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Step 5.5.3.1.5.1

Factor $2$ out of $- 12 x^{3} y^{2}$ .

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{2 (- 6 x^{3} y^{2})}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.5.2

Cancel the common factors.

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Step 5.5.3.1.5.2.1

Factor $2$ out of $2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})$ .

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{2 (- 6 x^{3} y^{2})}{2 (y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}))} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.5.2.2

Cancel the common factor.

Step 5.5.3.1.5.2.3

Rewrite the expression.

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x^{3} y^{2}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.6

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

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Step 5.5.3.1.6.1

Factor $y$ out of $- 6 x^{3} y^{2}$ .

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{y (- 6 x^{3} y)}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.6.2

Cancel the common factors.

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Step 5.5.3.1.6.2.1

Cancel the common factor.

Step 5.5.3.1.6.2.2

Rewrite the expression.

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} + \frac{- 6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.7

Move the negative in front of the fraction.

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{- 6 x y^{4}}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.8

Cancel the common factor of $- 6$ and $2$ .

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Step 5.5.3.1.8.1

Factor $2$ out of $- 6 x y^{4}$ .

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{2 (- 3 x y^{4})}{2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.8.2

Cancel the common factors.

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Step 5.5.3.1.8.2.1

Factor $2$ out of $2 y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})$ .

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{2 (- 3 x y^{4})}{2 (y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}))}$

Step 5.5.3.1.8.2.2

Cancel the common factor.

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{2 (- 3 x y^{4})}{2 (y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}))}$

Step 5.5.3.1.8.2.3

Rewrite the expression.

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{- 3 x y^{4}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.9

Cancel the common factor of $y^{4}$ and $y$ .

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Step 5.5.3.1.9.1

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

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{y (- 3 x y^{3})}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.9.2

Cancel the common factors.

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Step 5.5.3.1.9.2.1

Cancel the common factor.

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{y (- 3 x y^{3})}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.1.9.2.2

Rewrite the expression.

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} + \frac{- 3 x y^{3}}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$

Step 5.5.3.1.10

Move the negative in front of the fraction.

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x y^{3}}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$

Step 5.5.3.2

To write $\frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$y' = \frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} \cdot \frac{y}{y} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x y^{3}}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$

Step 5.5.3.3

Write each expression with a common denominator of $(3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}) y$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.3.3.1

Multiply $\frac{8 y x}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$ by $\frac{y}{y}$ .

$y' = \frac{8 y x y}{(3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}) y} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x y^{3}}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$

Step 5.5.3.3.2

Reorder the factors of $(3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}) y$ .

$y' = \frac{8 y x y}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x y^{3}}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$

Step 5.5.3.4

Combine the numerators over the common denominator.

$y' = \frac{8 y x y - 3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x y^{3}}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$

Step 5.5.3.5

Simplify the numerator.

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Step 5.5.3.5.1

Factor $x$ out of $8 y x y - 3 x^{5}$ .

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Step 5.5.3.5.1.1

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

$y' = \frac{x (8 y \cdot y) - 3 x^{5}}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})} - \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}} - \frac{3 x y^{3}}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$

Step 5.5.3.5.1.2

Factor $x$ out of $- 3 x^{5}$ .

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

Step 5.5.3.5.1.3

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

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

Step 5.5.3.5.2

Multiply $y$ by $y$ by adding the exponents.

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Step 5.5.3.5.2.1

Move $y$ .

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

Step 5.5.3.5.2.2

Multiply $y$ by $y$ .

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

Step 5.5.3.6

To write $- \frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 5.5.3.7

Write each expression with a common denominator of $y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.3.7.1

Multiply $\frac{6 x^{3} y}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$ by $\frac{y}{y}$ .

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

Step 5.5.3.7.2

Reorder the factors of $(3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}) y$ .

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

Step 5.5.3.8

Combine the numerators over the common denominator.

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

Step 5.5.3.9

Simplify the numerator.

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Step 5.5.3.9.1

Factor $x$ out of $x (8 y^{2} - 3 x^{4}) - 6 x^{3} y \cdot y$ .

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Step 5.5.3.9.1.1

Factor $x$ out of $- 6 x^{3} y \cdot y$ .

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

Step 5.5.3.9.1.2

Factor $x$ out of $x (8 y^{2} - 3 x^{4}) + x (- 6 x^{2} y \cdot y)$ .

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

Step 5.5.3.9.2

Multiply $y$ by $y$ by adding the exponents.

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Step 5.5.3.9.2.1

Move $y$ .

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

Step 5.5.3.9.2.2

Multiply $y$ by $y$ .

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

Step 5.5.3.10

To write $- \frac{3 x y^{3}}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 5.5.3.11

Write each expression with a common denominator of $y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.3.11.1

Multiply $\frac{3 x y^{3}}{3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}}$ by $\frac{y}{y}$ .

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

Step 5.5.3.11.2

Reorder the factors of $(3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2}) y$ .

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

Step 5.5.3.12

Combine the numerators over the common denominator.

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

Step 5.5.3.13

Simplify the numerator.

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Step 5.5.3.13.1

Factor $x$ out of $x (8 y^{2} - 3 x^{4} - 6 x^{2} y^{2}) - 3 x y^{3} y$ .

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Step 5.5.3.13.1.1

Factor $x$ out of $- 3 x y^{3} y$ .

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

Step 5.5.3.13.1.2

Factor $x$ out of $x (8 y^{2} - 3 x^{4} - 6 x^{2} y^{2}) + x (- 3 y^{3} y)$ .

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

Step 5.5.3.13.2

Multiply $y^{3}$ by $y$ by adding the exponents.

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Step 5.5.3.13.2.1

Move $y$ .

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

Step 5.5.3.13.2.2

Multiply $y$ by $y^{3}$ .

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Step 5.5.3.13.2.2.1

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

$y' = \frac{x (8 y^{2} - 3 x^{4} - 6 x^{2} y^{2} - 3 (y^{1} y^{3}))}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.13.2.2.2

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

$y' = \frac{x (8 y^{2} - 3 x^{4} - 6 x^{2} y^{2} - 3 y^{1 + 3})}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$

Step 5.5.3.13.2.3

Add $1$ and $3$ .

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

Step 6

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

$\frac{d y}{d x} = \frac{x (8 y^{2} - 3 x^{4} - 6 x^{2} y^{2} - 3 y^{4})}{y (3 x^{4} + 6 y^{2} x^{2} + 3 y^{4} - 8 x^{2})}$