Calculus Examples

$\arctan (2 x^{2} y) = x + 4 x y^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\arctan (2 x^{2} y)) = \frac{d}{d x} (x + 4 x y^{2})$

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

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Step 2.1.1

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

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

Step 2.1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

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

Step 2.1.3

Replace all occurrences of $u$ with $2 x^{2} y$ .

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

Step 2.2

Differentiate using the Constant Multiple Rule.

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Step 2.2.1

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

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

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Apply the product rule to $2 (x^{2} y)$ .

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

Step 2.2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Combine $2$ and $\frac{1}{1 + 4 {(x^{2} y)}^{2}}$ .

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

Step 2.3

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 $g (x) = y$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Simplify.

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Step 2.7.1

Apply the product rule to $x^{2} y$ .

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

Step 2.7.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.7.2.1

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

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

Step 2.7.2.2

Multiply $2$ by $2$ .

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

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

Step 2.7.3

Reorder the factors of $\frac{2}{1 + 4 x^{4} y^{2}} (x^{2} y' + 2 y x)$ .

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

Step 2.7.4

Multiply $x^{2} y' + 2 y x$ by $\frac{2}{1 + 4 x^{4} y^{2}}$ .

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

Step 2.7.5

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

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Step 2.7.5.1

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

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

Step 2.7.5.2

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

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

Step 2.7.5.3

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

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

Step 2.7.6

Move $2$ to the left of $x (x y' + 2 y)$ .

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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 3.2.3.1

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

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.4

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

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

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.2.7

Multiply $y^{2}$ by $1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Multiply $2$ by $4$ .

$1 + 8 x y y' + 4 y^{2}$

Step 3.3.3

Reorder terms.

$4 y^{2} + 8 x y y' + 1$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by $1 + 4 x^{4} y^{2}$ .

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

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $\frac{2 x (x y' + 2 y)}{1 + 4 x^{4} y^{2}} (1 + 4 x^{4} y^{2})$ .

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Step 5.2.1.1.1

Cancel the common factor of $1 + 4 x^{4} y^{2}$ .

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Step 5.2.1.1.1.1

Cancel the common factor.

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

Step 5.2.1.1.1.2

Rewrite the expression.

$2 x (x y' + 2 y) = (4 y^{2} + 8 x y y' + 1) (1 + 4 x^{4} y^{2})$

Step 5.2.1.1.2

Apply the distributive property.

$2 x (x y') + 2 x (2 y) = (4 y^{2} + 8 x y y' + 1) (1 + 4 x^{4} y^{2})$

Step 5.2.1.1.3

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

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Step 5.2.1.1.3.1

Move $x$ .

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

Step 5.2.1.1.3.2

Multiply $x$ by $x$ .

$2 x^{2} y' + 2 x (2 y) = (4 y^{2} + 8 x y y' + 1) (1 + 4 x^{4} y^{2})$

Step 5.2.1.1.4

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.1.5

Multiply $2$ by $2$ .

$2 x^{2} y' + 4 x y = (4 y^{2} + 8 x y y' + 1) (1 + 4 x^{4} y^{2})$

Step 5.2.1.1.6

Move $x^{2}$ .

$2 y' x^{2} + 4 x y = (4 y^{2} + 8 x y y' + 1) (1 + 4 x^{4} y^{2})$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $(4 y^{2} + 8 x y y' + 1) (1 + 4 x^{4} y^{2})$ .

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Step 5.2.2.1.1

Expand $(4 y^{2} + 8 x y y' + 1) (1 + 4 x^{4} y^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 5.2.2.1.2

Simplify terms.

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Step 5.2.2.1.2.1

Simplify each term.

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Step 5.2.2.1.2.1.1

Multiply $4$ by $1$ .

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

Step 5.2.2.1.2.1.2

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

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Step 5.2.2.1.2.1.2.1

Move $y^{2}$ .

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

Step 5.2.2.1.2.1.2.2

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

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

Step 5.2.2.1.2.1.2.3

Add $2$ and $2$ .

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

Step 5.2.2.1.2.1.3

Rewrite using the commutative property of multiplication.

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

Step 5.2.2.1.2.1.4

Multiply $4$ by $4$ .

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

Step 5.2.2.1.2.1.5

Multiply $8$ by $1$ .

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

Step 5.2.2.1.2.1.6

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

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Step 5.2.2.1.2.1.6.1

Move $x^{4}$ .

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

Step 5.2.2.1.2.1.6.2

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

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Step 5.2.2.1.2.1.6.2.1

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

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

Step 5.2.2.1.2.1.6.2.2

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

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

Step 5.2.2.1.2.1.6.3

Add $4$ and $1$ .

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 y^{4} x^{4} + 8 x y y' + 8 x^{5} y y' (4 y^{2}) + 1 \cdot 1 + 1 (4 x^{4} y^{2})$

Step 5.2.2.1.2.1.7

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

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Step 5.2.2.1.2.1.7.1

Move $y^{2}$ .

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 y^{4} x^{4} + 8 x y y' + 8 x^{5} (y^{2} y) y' \cdot 4 + 1 \cdot 1 + 1 (4 x^{4} y^{2})$

Step 5.2.2.1.2.1.7.2

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

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Step 5.2.2.1.2.1.7.2.1

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

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 y^{4} x^{4} + 8 x y y' + 8 x^{5} (y^{2} y^{1}) y' \cdot 4 + 1 \cdot 1 + 1 (4 x^{4} y^{2})$

Step 5.2.2.1.2.1.7.2.2

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

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 y^{4} x^{4} + 8 x y y' + 8 x^{5} y^{2 + 1} y' \cdot 4 + 1 \cdot 1 + 1 (4 x^{4} y^{2})$

Step 5.2.2.1.2.1.7.3

Add $2$ and $1$ .

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 y^{4} x^{4} + 8 x y y' + 8 x^{5} y^{3} y' \cdot 4 + 1 \cdot 1 + 1 (4 x^{4} y^{2})$

Step 5.2.2.1.2.1.8

Multiply $4$ by $8$ .

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 y^{4} x^{4} + 8 x y y' + 32 x^{5} y^{3} y' + 1 \cdot 1 + 1 (4 x^{4} y^{2})$

Step 5.2.2.1.2.1.9

Multiply $1$ by $1$ .

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 y^{4} x^{4} + 8 x y y' + 32 x^{5} y^{3} y' + 1 + 1 (4 x^{4} y^{2})$

Step 5.2.2.1.2.1.10

Multiply $4 x^{4} y^{2}$ by $1$ .

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 y^{4} x^{4} + 8 x y y' + 32 x^{5} y^{3} y' + 1 + 4 x^{4} y^{2}$

Step 5.2.2.1.2.2

Simplify the expression.

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Step 5.2.2.1.2.2.1

Move $y^{4}$ .

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 x^{4} y^{4} + 8 x y y' + 32 x^{5} y^{3} y' + 1 + 4 x^{4} y^{2}$

Step 5.2.2.1.2.2.2

Move $y$ .

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 x^{4} y^{4} + 8 x y' y + 32 x^{5} y^{3} y' + 1 + 4 x^{4} y^{2}$

Step 5.2.2.1.2.2.3

Move $x$ .

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 x^{4} y^{4} + 8 y' x y + 32 x^{5} y^{3} y' + 1 + 4 x^{4} y^{2}$

Step 5.2.2.1.2.2.4

Move $y^{3}$ .

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 x^{4} y^{4} + 8 y' x y + 32 x^{5} y' y^{3} + 1 + 4 x^{4} y^{2}$

Step 5.2.2.1.2.2.5

Move $x^{5}$ .

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 x^{4} y^{4} + 8 y' x y + 32 y' x^{5} y^{3} + 1 + 4 x^{4} y^{2}$

Step 5.2.2.1.2.2.6

Move $1$ .

$2 y' x^{2} + 4 x y = 4 y^{2} + 16 x^{4} y^{4} + 8 y' x y + 32 y' x^{5} y^{3} + 4 x^{4} y^{2} + 1$

Step 5.2.2.1.2.2.7

Move $4 y^{2}$ .

$2 y' x^{2} + 4 x y = 16 x^{4} y^{4} + 8 y' x y + 32 y' x^{5} y^{3} + 4 x^{4} y^{2} + 4 y^{2} + 1$

Step 5.2.2.1.2.2.8

Move $8 y' x y$ .

$2 y' x^{2} + 4 x y = 16 x^{4} y^{4} + 32 y' x^{5} y^{3} + 4 x^{4} y^{2} + 8 y' x y + 4 y^{2} + 1$

Step 5.2.2.1.2.2.9

Reorder $16 x^{4} y^{4}$ and $32 y' x^{5} y^{3}$ .

$2 y' x^{2} + 4 x y = 32 y' x^{5} y^{3} + 16 x^{4} y^{4} + 4 x^{4} y^{2} + 8 y' x y + 4 y^{2} + 1$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

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

$32 y' x^{5} y^{3} + 16 x^{4} y^{4} + 4 x^{4} y^{2} + 8 y' x y + 4 y^{2} + 1 = 2 y' x^{2} + 4 x y$

Step 5.3.2

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

$32 y' x^{5} y^{3} + 16 x^{4} y^{4} + 4 x^{4} y^{2} + 8 y' x y + 4 y^{2} + 1 - 2 y' x^{2} = 4 x y$

Step 5.3.3

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

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Step 5.3.3.1

Subtract $16 x^{4} y^{4}$ from both sides of the equation.

$32 y' x^{5} y^{3} + 4 x^{4} y^{2} + 8 y' x y + 4 y^{2} + 1 - 2 y' x^{2} = 4 x y - 16 x^{4} y^{4}$

Step 5.3.3.2

Subtract $4 x^{4} y^{2}$ from both sides of the equation.

$32 y' x^{5} y^{3} + 8 y' x y + 4 y^{2} + 1 - 2 y' x^{2} = 4 x y - 16 x^{4} y^{4} - 4 x^{4} y^{2}$

Step 5.3.3.3

Subtract $4 y^{2}$ from both sides of the equation.

$32 y' x^{5} y^{3} + 8 y' x y + 1 - 2 y' x^{2} = 4 x y - 16 x^{4} y^{4} - 4 x^{4} y^{2} - 4 y^{2}$

Step 5.3.3.4

Subtract $1$ from both sides of the equation.

$32 y' x^{5} y^{3} + 8 y' x y - 2 y' x^{2} = 4 x y - 16 x^{4} y^{4} - 4 x^{4} y^{2} - 4 y^{2} - 1$

Step 5.3.4

Factor $2 y' x$ out of $32 y' x^{5} y^{3} + 8 y' x y - 2 y' x^{2}$ .

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Step 5.3.4.1

Factor $2 y' x$ out of $32 y' x^{5} y^{3}$ .

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

Step 5.3.4.2

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

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

Step 5.3.4.3

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

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

Step 5.3.4.4

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

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

Step 5.3.4.5

Factor $2 y' x$ out of $2 y' x (16 x^{4} y^{3} + 4 y) + 2 y' x (- 1 x)$ .

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

Step 5.3.5

Rewrite $- 1 x$ as $- x$ .

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

Step 5.3.6

Divide each term in $2 y' x (16 x^{4} y^{3} + 4 y - x) = 4 x y - 16 x^{4} y^{4} - 4 x^{4} y^{2} - 4 y^{2} - 1$ by $2 x (16 x^{4} y^{3} + 4 y - x)$ and simplify.

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Step 5.3.6.1

Divide each term in $2 y' x (16 x^{4} y^{3} + 4 y - x) = 4 x y - 16 x^{4} y^{4} - 4 x^{4} y^{2} - 4 y^{2} - 1$ by $2 x (16 x^{4} y^{3} + 4 y - x)$ .

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

Step 5.3.6.2

Simplify the left side.

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Step 5.3.6.2.1

Cancel the common factor of $2$ .

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Step 5.3.6.2.1.1

Cancel the common factor.

Step 5.3.6.2.1.2

Rewrite the expression.

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

Step 5.3.6.2.2

Cancel the common factor of $x$ .

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Step 5.3.6.2.2.1

Cancel the common factor.

Step 5.3.6.2.2.2

Rewrite the expression.

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

Step 5.3.6.2.3

Cancel the common factor of $16 x^{4} y^{3} + 4 y - x$ .

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Step 5.3.6.2.3.1

Cancel the common factor.

Step 5.3.6.2.3.2

Divide $y'$ by $1$ .

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

Step 5.3.6.3

Simplify the right side.

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Step 5.3.6.3.1

Simplify each term.

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Step 5.3.6.3.1.1

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

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Step 5.3.6.3.1.1.1

Factor $2$ out of $4 x y$ .

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

Step 5.3.6.3.1.1.2

Cancel the common factors.

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Step 5.3.6.3.1.1.2.1

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

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

Step 5.3.6.3.1.1.2.2

Cancel the common factor.

Step 5.3.6.3.1.1.2.3

Rewrite the expression.

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

Step 5.3.6.3.1.2

Cancel the common factor of $x$ .

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Step 5.3.6.3.1.2.1

Cancel the common factor.

Step 5.3.6.3.1.2.2

Rewrite the expression.

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

Step 5.3.6.3.1.3

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

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Step 5.3.6.3.1.3.1

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

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

Step 5.3.6.3.1.3.2

Cancel the common factors.

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Step 5.3.6.3.1.3.2.1

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

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

Step 5.3.6.3.1.3.2.2

Cancel the common factor.

Step 5.3.6.3.1.3.2.3

Rewrite the expression.

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

Step 5.3.6.3.1.4

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

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Step 5.3.6.3.1.4.1

Factor $x$ out of $- 8 x^{4} y^{4}$ .

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

Step 5.3.6.3.1.4.2

Cancel the common factors.

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Step 5.3.6.3.1.4.2.1

Cancel the common factor.

Step 5.3.6.3.1.4.2.2

Rewrite the expression.

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

Step 5.3.6.3.1.5

Move the negative in front of the fraction.

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

Step 5.3.6.3.1.6

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

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Step 5.3.6.3.1.6.1

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

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

Step 5.3.6.3.1.6.2

Cancel the common factors.

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Step 5.3.6.3.1.6.2.1

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

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

Step 5.3.6.3.1.6.2.2

Cancel the common factor.

Step 5.3.6.3.1.6.2.3

Rewrite the expression.

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

Step 5.3.6.3.1.7

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

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Step 5.3.6.3.1.7.1

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

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

Step 5.3.6.3.1.7.2

Cancel the common factors.

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Step 5.3.6.3.1.7.2.1

Cancel the common factor.

Step 5.3.6.3.1.7.2.2

Rewrite the expression.

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

Step 5.3.6.3.1.8

Move the negative in front of the fraction.

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

Step 5.3.6.3.1.9

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

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Step 5.3.6.3.1.9.1

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

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

Step 5.3.6.3.1.9.2

Cancel the common factors.

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Step 5.3.6.3.1.9.2.1

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

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

Step 5.3.6.3.1.9.2.2

Cancel the common factor.

Step 5.3.6.3.1.9.2.3

Rewrite the expression.

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

Step 5.3.6.3.1.10

Move the negative in front of the fraction.

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

Step 5.3.6.3.1.11

Move the negative in front of the fraction.

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

Step 5.3.6.3.2

Simplify terms.

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Step 5.3.6.3.2.1

Combine the numerators over the common denominator.

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

Step 5.3.6.3.2.2

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

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Step 5.3.6.3.2.2.1

Factor $2 y$ out of $2 y$ .

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

Step 5.3.6.3.2.2.2

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

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

Step 5.3.6.3.2.2.3

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

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

Step 5.3.6.3.2.3

Combine the numerators over the common denominator.

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

Step 5.3.6.3.2.4

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

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Step 5.3.6.3.2.4.1

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

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

Step 5.3.6.3.2.4.2

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

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

Step 5.3.6.3.3

To write $\frac{2 y (1 - 4 x^{3} y^{3} - x^{3} y)}{16 x^{4} y^{3} + 4 y - x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 5.3.6.3.4

Write each expression with a common denominator of $(16 x^{4} y^{3} + 4 y - x) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.6.3.4.1

Multiply $\frac{2 y (1 - 4 x^{3} y^{3} - x^{3} y)}{16 x^{4} y^{3} + 4 y - x}$ by $\frac{x}{x}$ .

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

Step 5.3.6.3.4.2

Reorder the factors of $(16 x^{4} y^{3} + 4 y - x) x$ .

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

Step 5.3.6.3.5

Combine the numerators over the common denominator.

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

Step 5.3.6.3.6

Simplify the numerator.

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Step 5.3.6.3.6.1

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

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Step 5.3.6.3.6.1.1

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

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

Step 5.3.6.3.6.1.2

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

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

Step 5.3.6.3.6.1.3

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

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

Step 5.3.6.3.6.2

Apply the distributive property.

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

Step 5.3.6.3.6.3

Simplify.

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Step 5.3.6.3.6.3.1

Multiply $x$ by $1$ .

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

Step 5.3.6.3.6.3.2

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

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Step 5.3.6.3.6.3.2.1

Move $x$ .

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

Step 5.3.6.3.6.3.2.2

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

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Step 5.3.6.3.6.3.2.2.1

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

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

Step 5.3.6.3.6.3.2.2.2

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

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

Step 5.3.6.3.6.3.2.3

Add $1$ and $3$ .

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

Step 5.3.6.3.6.3.3

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

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Step 5.3.6.3.6.3.3.1

Move $x$ .

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

Step 5.3.6.3.6.3.3.2

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

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Step 5.3.6.3.6.3.3.2.1

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

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

Step 5.3.6.3.6.3.3.2.2

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

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

Step 5.3.6.3.6.3.3.3

Add $1$ and $3$ .

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

Step 5.3.6.3.7

To write $\frac{2 y (x - 4 x^{4} y^{3} - x^{4} y - y)}{x (16 x^{4} y^{3} + 4 y - x)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.3.6.3.8

Write each expression with a common denominator of $x (16 x^{4} y^{3} + 4 y - x) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.6.3.8.1

Multiply $\frac{2 y (x - 4 x^{4} y^{3} - x^{4} y - y)}{x (16 x^{4} y^{3} + 4 y - x)}$ by $\frac{2}{2}$ .

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

Step 5.3.6.3.8.2

Reorder the factors of $x (16 x^{4} y^{3} + 4 y - x) \cdot 2$ .

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

Step 5.3.6.3.9

Combine the numerators over the common denominator.

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

Step 5.3.6.3.10

Simplify the numerator.

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Step 5.3.6.3.10.1

Apply the distributive property.

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

Step 5.3.6.3.10.2

Simplify.

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Step 5.3.6.3.10.2.1

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

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Step 5.3.6.3.10.2.1.1

Move $y^{3}$ .

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

Step 5.3.6.3.10.2.1.2

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

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Step 5.3.6.3.10.2.1.2.1

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

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

Step 5.3.6.3.10.2.1.2.2

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

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

Step 5.3.6.3.10.2.1.3

Add $3$ and $1$ .

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

Step 5.3.6.3.10.2.2

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

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Step 5.3.6.3.10.2.2.1

Move $y$ .

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

Step 5.3.6.3.10.2.2.2

Multiply $y$ by $y$ .

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

Step 5.3.6.3.10.2.3

Rewrite using the commutative property of multiplication.

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

Step 5.3.6.3.10.3

Simplify each term.

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Step 5.3.6.3.10.3.1

Rewrite using the commutative property of multiplication.

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

Step 5.3.6.3.10.3.2

Multiply $2$ by $- 4$ .

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

Step 5.3.6.3.10.3.3

Rewrite using the commutative property of multiplication.

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

Step 5.3.6.3.10.3.4

Multiply $2$ by $- 1$ .

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

Step 5.3.6.3.10.3.5

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

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Step 5.3.6.3.10.3.5.1

Move $y$ .

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

Step 5.3.6.3.10.3.5.2

Multiply $y$ by $y$ .

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

Step 5.3.6.3.10.3.6

Multiply $2$ by $- 1$ .

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

Step 5.3.6.3.10.4

Apply the distributive property.

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

Step 5.3.6.3.10.5

Simplify.

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Step 5.3.6.3.10.5.1

Multiply $2$ by $2$ .

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

Step 5.3.6.3.10.5.2

Multiply $2$ by $- 8$ .

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

Step 5.3.6.3.10.5.3

Multiply $2$ by $- 2$ .

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

Step 5.3.6.3.10.5.4

Multiply $2$ by $- 2$ .

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

Step 6

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

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