Calculus Examples

${(x + y)}^{5} = y^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(x + y)}^{5}) = \frac{d}{d x} (y^{3})$

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 $f (x) = x^{5}$ and $g (x) = x + y$ .

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Step 2.1.1

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

$\frac{d}{d u} [u^{5}] \frac{d}{d x} [x + y]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$5 u^{4} \frac{d}{d x} [x + y]$

Step 2.1.3

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

$5 {(x + y)}^{4} \frac{d}{d x} [x + y]$

Step 2.2

Differentiate.

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Step 2.2.1

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

$5 {(x + y)}^{4} (\frac{d}{d x} [x] + \frac{d}{d x} [y])$

Step 2.2.2

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

$5 {(x + y)}^{4} (1 + \frac{d}{d x} [y])$

Step 2.3

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

$5 {(x + y)}^{4} (1 + y')$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

$3 y^{2} y'$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Simplify $5 {(x + y)}^{4} (1 + y')$ .

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Step 5.1.1

Rewrite.

$0 + 0 + 5 {(x + y)}^{4} (1 + y') = 3 y^{2} y'$

Step 5.1.2

Simplify by adding zeros.

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

Step 5.1.3

Apply the distributive property.

$5 {(x + y)}^{4} \cdot 1 + 5 {(x + y)}^{4} y' = 3 y^{2} y'$

Step 5.1.4

Simplify the expression.

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Step 5.1.4.1

Multiply $5$ by $1$ .

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

Step 5.1.4.2

Reorder factors in $5 {(x + y)}^{4} + 5 {(x + y)}^{4} y'$ .

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

Step 5.2

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

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Step 5.2.1

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

$5 {(x + y)}^{4} + 5 y' {(x + y)}^{4} - 3 y^{2} y' = 0$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Use the Binomial Theorem.

$5 (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}) + 5 y' {(x + y)}^{4} - 3 y^{2} y' = 0$

Step 5.2.2.2

Apply the distributive property.

$5 x^{4} + 5 (4 x^{3} y) + 5 (6 x^{2} y^{2}) + 5 (4 x y^{3}) + 5 y^{4} + 5 y' {(x + y)}^{4} - 3 y^{2} y' = 0$

Step 5.2.2.3

Simplify.

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Step 5.2.2.3.1

Multiply $4$ by $5$ .

$5 x^{4} + 20 (x^{3} y) + 5 (6 x^{2} y^{2}) + 5 (4 x y^{3}) + 5 y^{4} + 5 y' {(x + y)}^{4} - 3 y^{2} y' = 0$

Step 5.2.2.3.2

Multiply $6$ by $5$ .

$5 x^{4} + 20 (x^{3} y) + 30 (x^{2} y^{2}) + 5 (4 x y^{3}) + 5 y^{4} + 5 y' {(x + y)}^{4} - 3 y^{2} y' = 0$

Step 5.2.2.3.3

Multiply $4$ by $5$ .

$5 x^{4} + 20 (x^{3} y) + 30 (x^{2} y^{2}) + 20 (x y^{3}) + 5 y^{4} + 5 y' {(x + y)}^{4} - 3 y^{2} y' = 0$

Step 5.2.2.4

Remove parentheses.

$5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} + 5 y' {(x + y)}^{4} - 3 y^{2} y' = 0$

Step 5.2.2.5

Use the Binomial Theorem.

$5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} + 5 y' (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}) - 3 y^{2} y' = 0$

Step 5.2.2.6

Apply the distributive property.

$5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} + 5 y' x^{4} + 5 y' (4 x^{3} y) + 5 y' (6 x^{2} y^{2}) + 5 y' (4 x y^{3}) + 5 y' y^{4} - 3 y^{2} y' = 0$

Step 5.2.2.7

Simplify.

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Step 5.2.2.7.1

Multiply $4$ by $5$ .

$5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} + 5 y' x^{4} + 20 y' (x^{3} y) + 5 y' (6 x^{2} y^{2}) + 5 y' (4 x y^{3}) + 5 y' y^{4} - 3 y^{2} y' = 0$

Step 5.2.2.7.2

Multiply $6$ by $5$ .

$5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} + 5 y' x^{4} + 20 y' (x^{3} y) + 30 y' (x^{2} y^{2}) + 5 y' (4 x y^{3}) + 5 y' y^{4} - 3 y^{2} y' = 0$

Step 5.2.2.7.3

Multiply $4$ by $5$ .

$5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} + 5 y' x^{4} + 20 y' (x^{3} y) + 30 y' (x^{2} y^{2}) + 20 y' (x y^{3}) + 5 y' y^{4} - 3 y^{2} y' = 0$

Step 5.2.2.8

Remove parentheses.

$5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} + 5 y' x^{4} + 20 y' x^{3} y + 30 y' x^{2} y^{2} + 20 y' x y^{3} + 5 y' y^{4} - 3 y^{2} y' = 0$

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 $5 x^{4}$ from both sides of the equation.

$20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} + 5 y' x^{4} + 20 y' x^{3} y + 30 y' x^{2} y^{2} + 20 y' x y^{3} + 5 y' y^{4} - 3 y^{2} y' = - 5 x^{4}$

Step 5.3.2

Subtract $20 x^{3} y$ from both sides of the equation.

$30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} + 5 y' x^{4} + 20 y' x^{3} y + 30 y' x^{2} y^{2} + 20 y' x y^{3} + 5 y' y^{4} - 3 y^{2} y' = - 5 x^{4} - 20 x^{3} y$

Step 5.3.3

Subtract $30 x^{2} y^{2}$ from both sides of the equation.

$20 x y^{3} + 5 y^{4} + 5 y' x^{4} + 20 y' x^{3} y + 30 y' x^{2} y^{2} + 20 y' x y^{3} + 5 y' y^{4} - 3 y^{2} y' = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2}$

Step 5.3.4

Subtract $20 x y^{3}$ from both sides of the equation.

$5 y^{4} + 5 y' x^{4} + 20 y' x^{3} y + 30 y' x^{2} y^{2} + 20 y' x y^{3} + 5 y' y^{4} - 3 y^{2} y' = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3}$

Step 5.3.5

Subtract $5 y^{4}$ from both sides of the equation.

$5 y' x^{4} + 20 y' x^{3} y + 30 y' x^{2} y^{2} + 20 y' x y^{3} + 5 y' y^{4} - 3 y^{2} y' = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$

Step 5.4

Factor $y'$ out of $5 y' x^{4} + 20 y' x^{3} y + 30 y' x^{2} y^{2} + 20 y' x y^{3} + 5 y' y^{4} - 3 y^{2} y'$ .

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Step 5.4.1

Factor $y'$ out of $5 y' x^{4}$ .

$y' (5 x^{4}) + 20 y' x^{3} y + 30 y' x^{2} y^{2} + 20 y' x y^{3} + 5 y' y^{4} - 3 y^{2} y' = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$

Step 5.4.2

Factor $y'$ out of $20 y' x^{3} y$ .

$y' (5 x^{4}) + y' (20 x^{3} y) + 30 y' x^{2} y^{2} + 20 y' x y^{3} + 5 y' y^{4} - 3 y^{2} y' = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$

Step 5.4.3

Factor $y'$ out of $30 y' x^{2} y^{2}$ .

$y' (5 x^{4}) + y' (20 x^{3} y) + y' (30 x^{2} y^{2}) + 20 y' x y^{3} + 5 y' y^{4} - 3 y^{2} y' = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$

Step 5.4.4

Factor $y'$ out of $20 y' x y^{3}$ .

$y' (5 x^{4}) + y' (20 x^{3} y) + y' (30 x^{2} y^{2}) + y' (20 x y^{3}) + 5 y' y^{4} - 3 y^{2} y' = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$

Step 5.4.5

Factor $y'$ out of $5 y' y^{4}$ .

$y' (5 x^{4}) + y' (20 x^{3} y) + y' (30 x^{2} y^{2}) + y' (20 x y^{3}) + y' (5 y^{4}) - 3 y^{2} y' = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$

Step 5.4.6

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

$y' (5 x^{4}) + y' (20 x^{3} y) + y' (30 x^{2} y^{2}) + y' (20 x y^{3}) + y' (5 y^{4}) + y' (- 3 y^{2}) = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$

Step 5.4.7

Factor $y'$ out of $y' (5 x^{4}) + y' (20 x^{3} y)$ .

$y' (5 x^{4} + 20 x^{3} y) + y' (30 x^{2} y^{2}) + y' (20 x y^{3}) + y' (5 y^{4}) + y' (- 3 y^{2}) = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$

Step 5.4.8

Factor $y'$ out of $y' (5 x^{4} + 20 x^{3} y) + y' (30 x^{2} y^{2})$ .

$y' (5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2}) + y' (20 x y^{3}) + y' (5 y^{4}) + y' (- 3 y^{2}) = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$

Step 5.4.9

Factor $y'$ out of $y' (5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2}) + y' (20 x y^{3})$ .

$y' (5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3}) + y' (5 y^{4}) + y' (- 3 y^{2}) = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$

Step 5.4.10

Factor $y'$ out of $y' (5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3}) + y' (5 y^{4})$ .

$y' (5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4}) + y' (- 3 y^{2}) = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$

Step 5.4.11

Factor $y'$ out of $y' (5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4}) + y' (- 3 y^{2})$ .

$y' (5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}) = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$

Step 5.5

Divide each term in $y' (5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}) = - 5 x^{4} - 20 x^{3} y - 30 x^{2} y^{2} - 20 x y^{3} - 5 y^{4}$ by $5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}$ and simplify.

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Step 5.5.1

$\frac{y' (5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} = \frac{- 5 x^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 20 x^{3} y}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 30 x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}$ .

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Step 5.5.2.1.1

Cancel the common factor.

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 5 x^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 20 x^{3} y}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 30 x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify terms.

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Step 5.5.3.1.1

Simplify each term.

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Step 5.5.3.1.1.1

Move the negative in front of the fraction.

$y' = - \frac{(5) x^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 20 x^{3} y}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 30 x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.1.1.2

Move the negative in front of the fraction.

$y' = - \frac{5 x^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{(20) x^{3} y}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 30 x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.1.1.3

Move the negative in front of the fraction.

$y' = - \frac{5 x^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x^{3} y}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{(30) x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.1.1.4

Move the negative in front of the fraction.

$y' = - \frac{5 x^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x^{3} y}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{30 x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{(20) x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} + \frac{- 5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.1.1.5

Move the negative in front of the fraction.

$y' = - \frac{5 x^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x^{3} y}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{30 x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.1.2

Simplify terms.

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Step 5.5.3.1.2.1

Combine the numerators over the common denominator.

$y' = \frac{- 5 x^{4} - 20 x^{3} y}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{30 x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.1.2.2

Factor $5 x^{3}$ out of $- 5 x^{4} - 20 x^{3} y$ .

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Step 5.5.3.1.2.2.1

Factor $5 x^{3}$ out of $- 5 x^{4}$ .

$y' = \frac{5 x^{3} (- x) - 20 x^{3} y}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{30 x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.1.2.2.2

Factor $5 x^{3}$ out of $- 20 x^{3} y$ .

$y' = \frac{5 x^{3} (- x) + 5 x^{3} (- 4 y)}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{30 x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.1.2.2.3

Factor $5 x^{3}$ out of $5 x^{3} (- x) + 5 x^{3} (- 4 y)$ .

$y' = \frac{5 x^{3} (- x - 4 y)}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{30 x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.1.2.3

Combine the numerators over the common denominator.

$y' = \frac{5 x^{3} (- x - 4 y) - 30 x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.2

Simplify the numerator.

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Step 5.5.3.2.1

Factor $5 x^{2}$ out of $5 x^{3} (- x - 4 y) - 30 x^{2} y^{2}$ .

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Step 5.5.3.2.1.1

Factor $5 x^{2}$ out of $5 x^{3} (- x - 4 y)$ .

$y' = \frac{5 x^{2} (x (- x - 4 y)) - 30 x^{2} y^{2}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.2.1.2

Factor $5 x^{2}$ out of $- 30 x^{2} y^{2}$ .

$y' = \frac{5 x^{2} (x (- x - 4 y)) + 5 x^{2} (- 6 y^{2})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.2.1.3

Factor $5 x^{2}$ out of $5 x^{2} (x (- x - 4 y)) + 5 x^{2} (- 6 y^{2})$ .

$y' = \frac{5 x^{2} (x (- x - 4 y) - 6 y^{2})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.2.2

Apply the distributive property.

$y' = \frac{5 x^{2} (x (- x) + x (- 4 y) - 6 y^{2})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.2.3

Rewrite using the commutative property of multiplication.

$y' = \frac{5 x^{2} (- x \cdot x + x (- 4 y) - 6 y^{2})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.2.4

Rewrite using the commutative property of multiplication.

$y' = \frac{5 x^{2} (- x \cdot x - 4 x y - 6 y^{2})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.2.5

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

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Step 5.5.3.2.5.1

Move $x$ .

$y' = \frac{5 x^{2} (- (x \cdot x) - 4 x y - 6 y^{2})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.2.5.2

Multiply $x$ by $x$ .

$y' = \frac{5 x^{2} (- x^{2} - 4 x y - 6 y^{2})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.3

Combine the numerators over the common denominator.

$y' = \frac{5 x^{2} (- x^{2} - 4 x y - 6 y^{2}) - 20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4

Simplify the numerator.

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Step 5.5.3.4.1

Factor $5 x$ out of $5 x^{2} (- x^{2} - 4 x y - 6 y^{2}) - 20 x y^{3}$ .

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Step 5.5.3.4.1.1

Factor $5 x$ out of $5 x^{2} (- x^{2} - 4 x y - 6 y^{2})$ .

$y' = \frac{5 x (x (- x^{2} - 4 x y - 6 y^{2})) - 20 x y^{3}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4.1.2

Factor $5 x$ out of $- 20 x y^{3}$ .

$y' = \frac{5 x (x (- x^{2} - 4 x y - 6 y^{2})) + 5 x (- 4 y^{3})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4.1.3

Factor $5 x$ out of $5 x (x (- x^{2} - 4 x y - 6 y^{2})) + 5 x (- 4 y^{3})$ .

$y' = \frac{5 x (x (- x^{2} - 4 x y - 6 y^{2}) - 4 y^{3})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4.2

Apply the distributive property.

$y' = \frac{5 x (x (- x^{2}) + x (- 4 x y) + x (- 6 y^{2}) - 4 y^{3})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4.3

Simplify.

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Step 5.5.3.4.3.1

Rewrite using the commutative property of multiplication.

$y' = \frac{5 x (- x \cdot x^{2} + x (- 4 x y) + x (- 6 y^{2}) - 4 y^{3})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4.3.2

Rewrite using the commutative property of multiplication.

$y' = \frac{5 x (- x \cdot x^{2} - 4 x (x y) + x (- 6 y^{2}) - 4 y^{3})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4.3.3

Rewrite using the commutative property of multiplication.

$y' = \frac{5 x (- x \cdot x^{2} - 4 x (x y) - 6 x y^{2} - 4 y^{3})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4.4

Simplify each term.

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Step 5.5.3.4.4.1

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

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Step 5.5.3.4.4.1.1

Move $x^{2}$ .

$y' = \frac{5 x (- (x^{2} x) - 4 x (x y) - 6 x y^{2} - 4 y^{3})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4.4.1.2

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

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Step 5.5.3.4.4.1.2.1

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

$y' = \frac{5 x (- (x^{2} x^{1}) - 4 x (x y) - 6 x y^{2} - 4 y^{3})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4.4.1.2.2

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

$y' = \frac{5 x (- x^{2 + 1} - 4 x (x y) - 6 x y^{2} - 4 y^{3})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4.4.1.3

Add $2$ and $1$ .

$y' = \frac{5 x (- x^{3} - 4 x (x y) - 6 x y^{2} - 4 y^{3})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4.4.2

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

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Step 5.5.3.4.4.2.1

Move $x$ .

$y' = \frac{5 x (- x^{3} - 4 (x \cdot x) y - 6 x y^{2} - 4 y^{3})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.4.4.2.2

Multiply $x$ by $x$ .

$y' = \frac{5 x (- x^{3} - 4 x^{2} y - 6 x y^{2} - 4 y^{3})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}} - \frac{5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.5

Combine the numerators over the common denominator.

$y' = \frac{5 x (- x^{3} - 4 x^{2} y - 6 x y^{2} - 4 y^{3}) - 5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6

Simplify the numerator.

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Step 5.5.3.6.1

Factor $5$ out of $5 x (- x^{3} - 4 x^{2} y - 6 x y^{2} - 4 y^{3}) - 5 y^{4}$ .

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Step 5.5.3.6.1.1

Factor $5$ out of $5 x (- x^{3} - 4 x^{2} y - 6 x y^{2} - 4 y^{3})$ .

$y' = \frac{5 (x (- x^{3} - 4 x^{2} y - 6 x y^{2} - 4 y^{3})) - 5 y^{4}}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.1.2

Factor $5$ out of $- 5 y^{4}$ .

$y' = \frac{5 (x (- x^{3} - 4 x^{2} y - 6 x y^{2} - 4 y^{3})) + 5 (- y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.1.3

Factor $5$ out of $5 (x (- x^{3} - 4 x^{2} y - 6 x y^{2} - 4 y^{3})) + 5 (- y^{4})$ .

$y' = \frac{5 (x (- x^{3} - 4 x^{2} y - 6 x y^{2} - 4 y^{3}) - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.2

Apply the distributive property.

$y' = \frac{5 (x (- x^{3}) + x (- 4 x^{2} y) + x (- 6 x y^{2}) + x (- 4 y^{3}) - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.3

Simplify.

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Step 5.5.3.6.3.1

Rewrite using the commutative property of multiplication.

$y' = \frac{5 (- x \cdot x^{3} + x (- 4 x^{2} y) + x (- 6 x y^{2}) + x (- 4 y^{3}) - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.3.2

Rewrite using the commutative property of multiplication.

$y' = \frac{5 (- x \cdot x^{3} - 4 x (x^{2} y) + x (- 6 x y^{2}) + x (- 4 y^{3}) - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.3.3

Rewrite using the commutative property of multiplication.

$y' = \frac{5 (- x \cdot x^{3} - 4 x (x^{2} y) - 6 x (x y^{2}) + x (- 4 y^{3}) - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.3.4

Rewrite using the commutative property of multiplication.

$y' = \frac{5 (- x \cdot x^{3} - 4 x (x^{2} y) - 6 x (x y^{2}) - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.4

Simplify each term.

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Step 5.5.3.6.4.1

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

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Step 5.5.3.6.4.1.1

Move $x^{3}$ .

$y' = \frac{5 (- (x^{3} x) - 4 x (x^{2} y) - 6 x (x y^{2}) - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.4.1.2

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

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Step 5.5.3.6.4.1.2.1

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

$y' = \frac{5 (- (x^{3} x^{1}) - 4 x (x^{2} y) - 6 x (x y^{2}) - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.4.1.2.2

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

$y' = \frac{5 (- x^{3 + 1} - 4 x (x^{2} y) - 6 x (x y^{2}) - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.4.1.3

Add $3$ and $1$ .

$y' = \frac{5 (- x^{4} - 4 x (x^{2} y) - 6 x (x y^{2}) - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.4.2

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

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Step 5.5.3.6.4.2.1

Move $x^{2}$ .

$y' = \frac{5 (- x^{4} - 4 (x^{2} x) y - 6 x (x y^{2}) - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.4.2.2

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

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Step 5.5.3.6.4.2.2.1

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

$y' = \frac{5 (- x^{4} - 4 (x^{2} x^{1}) y - 6 x (x y^{2}) - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.4.2.2.2

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

$y' = \frac{5 (- x^{4} - 4 x^{2 + 1} y - 6 x (x y^{2}) - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.4.2.3

Add $2$ and $1$ .

$y' = \frac{5 (- x^{4} - 4 x^{3} y - 6 x (x y^{2}) - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.4.3

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

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Step 5.5.3.6.4.3.1

Move $x$ .

$y' = \frac{5 (- x^{4} - 4 x^{3} y - 6 (x \cdot x) y^{2} - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.6.4.3.2

Multiply $x$ by $x$ .

$y' = \frac{5 (- x^{4} - 4 x^{3} y - 6 x^{2} y^{2} - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.7

Simplify with factoring out.

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Step 5.5.3.7.1

Factor $- 1$ out of $- x^{4}$ .

$y' = \frac{5 (- (x^{4}) - 4 x^{3} y - 6 x^{2} y^{2} - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.7.2

Factor $- 1$ out of $- 4 x^{3} y$ .

$y' = \frac{5 (- (x^{4}) - (4 x^{3} y) - 6 x^{2} y^{2} - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.7.3

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

$y' = \frac{5 (- (x^{4} + 4 x^{3} y) - 6 x^{2} y^{2} - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.7.4

Factor $- 1$ out of $- 6 x^{2} y^{2}$ .

$y' = \frac{5 (- (x^{4} + 4 x^{3} y) - (6 x^{2} y^{2}) - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.7.5

Factor $- 1$ out of $- (x^{4} + 4 x^{3} y) - (6 x^{2} y^{2})$ .

$y' = \frac{5 (- (x^{4} + 4 x^{3} y + 6 x^{2} y^{2}) - 4 x y^{3} - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.7.6

Factor $- 1$ out of $- 4 x y^{3}$ .

$y' = \frac{5 (- (x^{4} + 4 x^{3} y + 6 x^{2} y^{2}) - (4 x y^{3}) - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.7.7

Factor $- 1$ out of $- (x^{4} + 4 x^{3} y + 6 x^{2} y^{2}) - (4 x y^{3})$ .

$y' = \frac{5 (- (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3}) - y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.7.8

Factor $- 1$ out of $- y^{4}$ .

$y' = \frac{5 (- (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3}) - (y^{4}))}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.7.9

Factor $- 1$ out of $- (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3}) - (y^{4})$ .

$y' = \frac{5 (- (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}))}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.7.10

Simplify the expression.

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Step 5.5.3.7.10.1

Rewrite $- (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4})$ as $- 1 (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4})$ .

$y' = \frac{5 (- 1 (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}))}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 5.5.3.7.10.2

Move the negative in front of the fraction.

$y' = - \frac{5 (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$

Step 6

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

$\frac{d y}{d x} = - \frac{5 (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4})}{5 x^{4} + 20 x^{3} y + 30 x^{2} y^{2} + 20 x y^{3} + 5 y^{4} - 3 y^{2}}$