Find dy/dx arctan(4x^2y)=x+5xy^2

$\arctan (4 x^{2} y) = x + 5 x y^{2}$

Differentiate both sides of the equation.

$\frac{d}{d x} (\arctan (4 x^{2} y)) = \frac{d}{d x} (x + 5 x y^{2})$

Differentiate the left side of the equation.

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 $g (x) = 4 x^{2} y$ .

To apply the Chain Rule, set $u$ as $4 x^{2} y$ .

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

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

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

Replace all occurrences of $u$ with $4 x^{2} y$ .

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

Differentiate using the Constant Multiple Rule.

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

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

Simplify the expression.

Apply the product rule to $4 (x^{2} y)$ .

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

Raise $4$ to the power of $2$ .

$\frac{1}{1 + 16 {(x^{2} y)}^{2}} \frac{d}{d x} [4 x^{2} y]$

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

$\frac{1}{1 + 16 {(x^{2} y)}^{2}} (4 \frac{d}{d x} [x^{2} y])$

Combine $4$ and $\frac{1}{1 + 16 {(x^{2} y)}^{2}}$ .

$\frac{4}{1 + 16 {(x^{2} y)}^{2}} \frac{d}{d x} [x^{2} y]$

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 $g (x) = y$ .

$\frac{4}{1 + 16 {(x^{2} y)}^{2}} (x^{2} \frac{d}{d x} [y] + y \frac{d}{d x} [x^{2}])$

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

$\frac{4}{1 + 16 {(x^{2} y)}^{2}} (x^{2} y' + y \frac{d}{d x} [x^{2}])$

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

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

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

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

Simplify.

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

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

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

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

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

Multiply $2$ by $2$ .

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

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

Reorder the factors of $\frac{4}{1 + 16 x^{4} y^{2}} (x^{2} y' + 2 y x)$ .

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

Multiply $x^{2} y' + 2 y x$ by $\frac{4}{1 + 16 x^{4} y^{2}}$ .

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

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

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

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

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

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

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

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

Move $4$ to the left of $x (x y' + 2 y)$ .

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

Differentiate the right side of the equation.

Differentiate.

By the Sum Rule, the derivative of $x + 5 x y^{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5 x y^{2}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [5 x y^{2}]$

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

$1 + \frac{d}{d x} [5 x y^{2}]$

Evaluate $\frac{d}{d x} [5 x y^{2}]$ .

Since $5$ is constant with respect to $x$ , the derivative of $5 x y^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x y^{2}]$ .

$1 + 5 \frac{d}{d x} [x y^{2}]$

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

$1 + 5 (x \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x])$

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 $g (x) = y$ .

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

$1 + 5 (x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

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

$1 + 5 (x (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

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

$1 + 5 (x (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

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

$1 + 5 (x (2 y y') + y^{2} \frac{d}{d x} [x])$

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

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

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

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

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

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

Simplify.

Apply the distributive property.

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

Multiply $2$ by $5$ .

$1 + 10 x y y' + 5 y^{2}$

Reorder terms.

$5 y^{2} + 10 x y y' + 1$

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

$\frac{4 x (x y' + 2 y)}{1 + 16 x^{4} y^{2}} = 5 y^{2} + 10 x y y' + 1$

Solve for $y'$ .

Multiply both sides by $1 + 16 x^{4} y^{2}$ .

$\frac{4 x (x y' + 2 y)}{1 + 16 x^{4} y^{2}} (1 + 16 x^{4} y^{2}) = (5 y^{2} + 10 x y y' + 1) (1 + 16 x^{4} y^{2})$

Simplify.

Simplify the left side.

Simplify $\frac{4 x (x y' + 2 y)}{1 + 16 x^{4} y^{2}} (1 + 16 x^{4} y^{2})$ .

Cancel the common factor of $1 + 16 x^{4} y^{2}$ .

Cancel the common factor.

$\frac{4 x (x y' + 2 y)}{1 + 16 x^{4} y^{2}} (1 + 16 x^{4} y^{2}) = (5 y^{2} + 10 x y y' + 1) (1 + 16 x^{4} y^{2})$

Rewrite the expression.

$4 x (x y' + 2 y) = (5 y^{2} + 10 x y y' + 1) (1 + 16 x^{4} y^{2})$

Apply the distributive property.

$4 x (x y') + 4 x (2 y) = (5 y^{2} + 10 x y y' + 1) (1 + 16 x^{4} y^{2})$

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

Move $x$ .

$4 (x \cdot x) y' + 4 x (2 y) = (5 y^{2} + 10 x y y' + 1) (1 + 16 x^{4} y^{2})$

Multiply $x$ by $x$ .

$4 x^{2} y' + 4 x (2 y) = (5 y^{2} + 10 x y y' + 1) (1 + 16 x^{4} y^{2})$

Rewrite using the commutative property of multiplication.

$4 x^{2} y' + 4 \cdot 2 x y = (5 y^{2} + 10 x y y' + 1) (1 + 16 x^{4} y^{2})$

Multiply $4$ by $2$ .

$4 x^{2} y' + 8 x y = (5 y^{2} + 10 x y y' + 1) (1 + 16 x^{4} y^{2})$

Move $x^{2}$ .

$4 y' x^{2} + 8 x y = (5 y^{2} + 10 x y y' + 1) (1 + 16 x^{4} y^{2})$

Simplify the right side.

Simplify $(5 y^{2} + 10 x y y' + 1) (1 + 16 x^{4} y^{2})$ .

Expand $(5 y^{2} + 10 x y y' + 1) (1 + 16 x^{4} y^{2})$ by multiplying each term in the first expression by each term in the second expression.

$4 y' x^{2} + 8 x y = 5 y^{2} \cdot 1 + 5 y^{2} (16 x^{4} y^{2}) + 10 x y y' \cdot 1 + 10 x y y' (16 x^{4} y^{2}) + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

Simplify terms.

Simplify each term.

Multiply $5$ by $1$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 5 y^{2} (16 x^{4} y^{2}) + 10 x y y' \cdot 1 + 10 x y y' (16 x^{4} y^{2}) + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

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

Move $y^{2}$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 5 (y^{2} y^{2}) (16 x^{4}) + 10 x y y' \cdot 1 + 10 x y y' (16 x^{4} y^{2}) + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

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

$4 y' x^{2} + 8 x y = 5 y^{2} + 5 y^{2 + 2} (16 x^{4}) + 10 x y y' \cdot 1 + 10 x y y' (16 x^{4} y^{2}) + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

Add $2$ and $2$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 5 y^{4} (16 x^{4}) + 10 x y y' \cdot 1 + 10 x y y' (16 x^{4} y^{2}) + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

Rewrite using the commutative property of multiplication.

$4 y' x^{2} + 8 x y = 5 y^{2} + 5 \cdot 16 y^{4} x^{4} + 10 x y y' \cdot 1 + 10 x y y' (16 x^{4} y^{2}) + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

Multiply $5$ by $16$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' \cdot 1 + 10 x y y' (16 x^{4} y^{2}) + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

Multiply $10$ by $1$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' + 10 x y y' (16 x^{4} y^{2}) + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

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

Move $x^{4}$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' + 10 (x^{4} x) y y' (16 y^{2}) + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

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

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

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' + 10 (x^{4} x^{1}) y y' (16 y^{2}) + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

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

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' + 10 x^{4 + 1} y y' (16 y^{2}) + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

Add $4$ and $1$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' + 10 x^{5} y y' (16 y^{2}) + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

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

Move $y^{2}$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' + 10 x^{5} (y^{2} y) y' \cdot 16 + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

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

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

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' + 10 x^{5} (y^{2} y^{1}) y' \cdot 16 + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

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

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' + 10 x^{5} y^{2 + 1} y' \cdot 16 + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

Add $2$ and $1$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' + 10 x^{5} y^{3} y' \cdot 16 + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

Multiply $16$ by $10$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' + 160 x^{5} y^{3} y' + 1 \cdot 1 + 1 (16 x^{4} y^{2})$

Multiply $1$ by $1$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' + 160 x^{5} y^{3} y' + 1 + 1 (16 x^{4} y^{2})$

Multiply $16 x^{4} y^{2}$ by $1$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 y^{4} x^{4} + 10 x y y' + 160 x^{5} y^{3} y' + 1 + 16 x^{4} y^{2}$

Simplify the expression.

Move $y^{4}$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 x^{4} y^{4} + 10 x y y' + 160 x^{5} y^{3} y' + 1 + 16 x^{4} y^{2}$

Move $y$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 x^{4} y^{4} + 10 x y' y + 160 x^{5} y^{3} y' + 1 + 16 x^{4} y^{2}$

Move $x$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 x^{4} y^{4} + 10 y' x y + 160 x^{5} y^{3} y' + 1 + 16 x^{4} y^{2}$

Move $y^{3}$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 x^{4} y^{4} + 10 y' x y + 160 x^{5} y' y^{3} + 1 + 16 x^{4} y^{2}$

Move $x^{5}$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 x^{4} y^{4} + 10 y' x y + 160 y' x^{5} y^{3} + 1 + 16 x^{4} y^{2}$

Move $1$ .

$4 y' x^{2} + 8 x y = 5 y^{2} + 80 x^{4} y^{4} + 10 y' x y + 160 y' x^{5} y^{3} + 16 x^{4} y^{2} + 1$

Move $5 y^{2}$ .

$4 y' x^{2} + 8 x y = 80 x^{4} y^{4} + 10 y' x y + 160 y' x^{5} y^{3} + 16 x^{4} y^{2} + 5 y^{2} + 1$

Move $10 y' x y$ .

$4 y' x^{2} + 8 x y = 80 x^{4} y^{4} + 160 y' x^{5} y^{3} + 16 x^{4} y^{2} + 10 y' x y + 5 y^{2} + 1$

Reorder $80 x^{4} y^{4}$ and $160 y' x^{5} y^{3}$ .

$4 y' x^{2} + 8 x y = 160 y' x^{5} y^{3} + 80 x^{4} y^{4} + 16 x^{4} y^{2} + 10 y' x y + 5 y^{2} + 1$

Solve for $y'$ .

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

$160 y' x^{5} y^{3} + 80 x^{4} y^{4} + 16 x^{4} y^{2} + 10 y' x y + 5 y^{2} + 1 = 4 y' x^{2} + 8 x y$

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

$160 y' x^{5} y^{3} + 80 x^{4} y^{4} + 16 x^{4} y^{2} + 10 y' x y + 5 y^{2} + 1 - 4 y' x^{2} = 8 x y$

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

Subtract $80 x^{4} y^{4}$ from both sides of the equation.

$160 y' x^{5} y^{3} + 16 x^{4} y^{2} + 10 y' x y + 5 y^{2} + 1 - 4 y' x^{2} = 8 x y - 80 x^{4} y^{4}$

Subtract $16 x^{4} y^{2}$ from both sides of the equation.

$160 y' x^{5} y^{3} + 10 y' x y + 5 y^{2} + 1 - 4 y' x^{2} = 8 x y - 80 x^{4} y^{4} - 16 x^{4} y^{2}$

Subtract $5 y^{2}$ from both sides of the equation.

$160 y' x^{5} y^{3} + 10 y' x y + 1 - 4 y' x^{2} = 8 x y - 80 x^{4} y^{4} - 16 x^{4} y^{2} - 5 y^{2}$

Subtract $1$ from both sides of the equation.

$160 y' x^{5} y^{3} + 10 y' x y - 4 y' x^{2} = 8 x y - 80 x^{4} y^{4} - 16 x^{4} y^{2} - 5 y^{2} - 1$

Factor $2 y' x$ out of $160 y' x^{5} y^{3} + 10 y' x y - 4 y' x^{2}$ .

Factor $2 y' x$ out of $160 y' x^{5} y^{3}$ .

$2 y' x (80 x^{4} y^{3}) + 10 y' x y - 4 y' x^{2} = 8 x y - 80 x^{4} y^{4} - 16 x^{4} y^{2} - 5 y^{2} - 1$

Factor $2 y' x$ out of $10 y' x y$ .

$2 y' x (80 x^{4} y^{3}) + 2 y' x (5 y) - 4 y' x^{2} = 8 x y - 80 x^{4} y^{4} - 16 x^{4} y^{2} - 5 y^{2} - 1$

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

$2 y' x (80 x^{4} y^{3}) + 2 y' x (5 y) + 2 y' x (- 2 x) = 8 x y - 80 x^{4} y^{4} - 16 x^{4} y^{2} - 5 y^{2} - 1$

Factor $2 y' x$ out of $2 y' x (80 x^{4} y^{3}) + 2 y' x (5 y)$ .

$2 y' x (80 x^{4} y^{3} + 5 y) + 2 y' x (- 2 x) = 8 x y - 80 x^{4} y^{4} - 16 x^{4} y^{2} - 5 y^{2} - 1$

Factor $2 y' x$ out of $2 y' x (80 x^{4} y^{3} + 5 y) + 2 y' x (- 2 x)$ .

$2 y' x (80 x^{4} y^{3} + 5 y - 2 x) = 8 x y - 80 x^{4} y^{4} - 16 x^{4} y^{2} - 5 y^{2} - 1$

Divide each term in $2 y' x (80 x^{4} y^{3} + 5 y - 2 x) = 8 x y - 80 x^{4} y^{4} - 16 x^{4} y^{2} - 5 y^{2} - 1$ by $2 x (80 x^{4} y^{3} + 5 y - 2 x)$ and simplify.

Divide each term in $2 y' x (80 x^{4} y^{3} + 5 y - 2 x) = 8 x y - 80 x^{4} y^{4} - 16 x^{4} y^{2} - 5 y^{2} - 1$ by $2 x (80 x^{4} y^{3} + 5 y - 2 x)$ .

$\frac{2 y' x (80 x^{4} y^{3} + 5 y - 2 x)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} = \frac{8 x y}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 80 x^{4} y^{4}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

Rewrite the expression.

$\frac{y' x (80 x^{4} y^{3} + 5 y - 2 x)}{x (80 x^{4} y^{3} + 5 y - 2 x)} = \frac{8 x y}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 80 x^{4} y^{4}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Cancel the common factor of $x$ .

Cancel the common factor.

Rewrite the expression.

$\frac{y' (80 x^{4} y^{3} + 5 y - 2 x)}{80 x^{4} y^{3} + 5 y - 2 x} = \frac{8 x y}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 80 x^{4} y^{4}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Cancel the common factor of $80 x^{4} y^{3} + 5 y - 2 x$ .

Cancel the common factor.

Divide $y'$ by $1$ .

$y' = \frac{8 x y}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 80 x^{4} y^{4}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $8$ and $2$ .

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

$y' = \frac{2 (4 x y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 80 x^{4} y^{4}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Cancel the common factors.

Factor $2$ out of $2 x (80 x^{4} y^{3} + 5 y - 2 x)$ .

$y' = \frac{2 (4 x y)}{2 (x (80 x^{4} y^{3} + 5 y - 2 x))} + \frac{- 80 x^{4} y^{4}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Cancel the common factor.

Rewrite the expression.

$y' = \frac{4 x y}{x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 80 x^{4} y^{4}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Cancel the common factor of $x$ .

Cancel the common factor.

Rewrite the expression.

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{- 80 x^{4} y^{4}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Cancel the common factor of $- 80$ and $2$ .

Factor $2$ out of $- 80 x^{4} y^{4}$ .

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{2 (- 40 x^{4} y^{4})}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Cancel the common factors.

Factor $2$ out of $2 x (80 x^{4} y^{3} + 5 y - 2 x)$ .

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{2 (- 40 x^{4} y^{4})}{2 (x (80 x^{4} y^{3} + 5 y - 2 x))} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Cancel the common factor.

Rewrite the expression.

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{- 40 x^{4} y^{4}}{x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

Factor $x$ out of $- 40 x^{4} y^{4}$ .

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{x (- 40 x^{3} y^{4})}{x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Cancel the common factors.

Cancel the common factor.

Rewrite the expression.

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{- 40 x^{3} y^{4}}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Move the negative in front of the fraction.

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{(40) x^{3} y^{4}}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{- 16 x^{4} y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

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

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{40 x^{3} y^{4}}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{2 (- 8 x^{4} y^{2})}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Cancel the common factors.

Factor $2$ out of $2 x (80 x^{4} y^{3} + 5 y - 2 x)$ .

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{40 x^{3} y^{4}}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{2 (- 8 x^{4} y^{2})}{2 (x (80 x^{4} y^{3} + 5 y - 2 x))} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Cancel the common factor.

Rewrite the expression.

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{40 x^{3} y^{4}}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{- 8 x^{4} y^{2}}{x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

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

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{40 x^{3} y^{4}}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{x (- 8 x^{3} y^{2})}{x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Cancel the common factors.

Cancel the common factor.

Rewrite the expression.

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{40 x^{3} y^{4}}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{- 8 x^{3} y^{2}}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Move the negative in front of the fraction.

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{40 x^{3} y^{4}}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{(8) x^{3} y^{2}}{80 x^{4} y^{3} + 5 y - 2 x} + \frac{- 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Move the negative in front of the fraction.

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{40 x^{3} y^{4}}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{8 x^{3} y^{2}}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{(5) y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} + \frac{- 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Move the negative in front of the fraction.

$y' = \frac{4 y}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{40 x^{3} y^{4}}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{8 x^{3} y^{2}}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Simplify terms.

Combine the numerators over the common denominator.

$y' = \frac{4 y - 40 x^{3} y^{4}}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{8 x^{3} y^{2}}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Factor $4 y$ out of $4 y - 40 x^{3} y^{4}$ .

Factor $4 y$ out of $4 y$ .

$y' = \frac{4 y (1) - 40 x^{3} y^{4}}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{8 x^{3} y^{2}}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Factor $4 y$ out of $- 40 x^{3} y^{4}$ .

$y' = \frac{4 y (1) + 4 y (- 10 x^{3} y^{3})}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{8 x^{3} y^{2}}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Factor $4 y$ out of $4 y (1) + 4 y (- 10 x^{3} y^{3})$ .

$y' = \frac{4 y (1 - 10 x^{3} y^{3})}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{8 x^{3} y^{2}}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Combine the numerators over the common denominator.

$y' = \frac{4 y (1 - 10 x^{3} y^{3}) - 8 x^{3} y^{2}}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Factor $4 y$ out of $4 y (1 - 10 x^{3} y^{3}) - 8 x^{3} y^{2}$ .

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

$y' = \frac{4 y (1 - 10 x^{3} y^{3}) + 4 y (- 2 x^{3} y)}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Factor $4 y$ out of $4 y (1 - 10 x^{3} y^{3}) + 4 y (- 2 x^{3} y)$ .

$y' = \frac{4 y (1 - 10 x^{3} y^{3} - 2 x^{3} y)}{80 x^{4} y^{3} + 5 y - 2 x} - \frac{5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

To write $\frac{4 y (1 - 10 x^{3} y^{3} - 2 x^{3} y)}{80 x^{4} y^{3} + 5 y - 2 x}$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$y' = \frac{4 y (1 - 10 x^{3} y^{3} - 2 x^{3} y)}{80 x^{4} y^{3} + 5 y - 2 x} \cdot \frac{2 x}{2 x} - \frac{5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Write each expression with a common denominator of $(80 x^{4} y^{3} + 5 y - 2 x) \cdot 2 x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4 y (1 - 10 x^{3} y^{3} - 2 x^{3} y)}{80 x^{4} y^{3} + 5 y - 2 x}$ by $\frac{2 x}{2 x}$ .

$y' = \frac{4 y (1 - 10 x^{3} y^{3} - 2 x^{3} y) (2 x)}{(80 x^{4} y^{3} + 5 y - 2 x) (2 x)} - \frac{5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Reorder the factors of $(80 x^{4} y^{3} + 5 y - 2 x) (2 x)$ .

$y' = \frac{4 y (1 - 10 x^{3} y^{3} - 2 x^{3} y) (2 x)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Combine the numerators over the common denominator.

$y' = \frac{4 y (1 - 10 x^{3} y^{3} - 2 x^{3} y) (2 x) - 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Simplify the numerator.

Factor $y$ out of $4 y (1 - 10 x^{3} y^{3} - 2 x^{3} y) (2 x) - 5 y^{2}$ .

Factor $y$ out of $4 y (1 - 10 x^{3} y^{3} - 2 x^{3} y) (2 x)$ .

$y' = \frac{y ((4 (1 - 10 x^{3} y^{3} - 2 x^{3} y)) (2 x)) - 5 y^{2}}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

$y' = \frac{y ((4 (1 - 10 x^{3} y^{3} - 2 x^{3} y)) (2 x)) + y (- 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Factor $y$ out of $y ((4 (1 - 10 x^{3} y^{3} - 2 x^{3} y)) (2 x)) + y (- 5 y)$ .

$y' = \frac{y ((4 (1 - 10 x^{3} y^{3} - 2 x^{3} y)) (2 x) - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Apply the distributive property.

$y' = \frac{y ((4 \cdot 1 + 4 (- 10 x^{3} y^{3}) + 4 (- 2 x^{3} y)) (2 x) - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Simplify.

Multiply $4$ by $1$ .

$y' = \frac{y ((4 + 4 (- 10 x^{3} y^{3}) + 4 (- 2 x^{3} y)) (2 x) - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Multiply $- 10$ by $4$ .

$y' = \frac{y ((4 - 40 (x^{3} y^{3}) + 4 (- 2 x^{3} y)) (2 x) - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Multiply $- 2$ by $4$ .

$y' = \frac{y ((4 - 40 (x^{3} y^{3}) - 8 (x^{3} y)) (2 x) - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Remove parentheses.

$y' = \frac{y ((4 - 40 x^{3} y^{3} - 8 x^{3} y) (2 x) - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Apply the distributive property.

$y' = \frac{y (4 (2 x) - 40 x^{3} y^{3} (2 x) - 8 x^{3} y (2 x) - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Simplify.

Multiply $2$ by $4$ .

$y' = \frac{y (8 x - 40 x^{3} y^{3} (2 x) - 8 x^{3} y (2 x) - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

Move $x$ .

$y' = \frac{y (8 x - 40 (x \cdot x^{3}) y^{3} \cdot 2 - 8 x^{3} y (2 x) - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

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

$y' = \frac{y (8 x - 40 (x^{1} x^{3}) y^{3} \cdot 2 - 8 x^{3} y (2 x) - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

$y' = \frac{y (8 x - 40 x^{1 + 3} y^{3} \cdot 2 - 8 x^{3} y (2 x) - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Add $1$ and $3$ .

$y' = \frac{y (8 x - 40 x^{4} y^{3} \cdot 2 - 8 x^{3} y (2 x) - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

Move $x$ .

$y' = \frac{y (8 x - 40 x^{4} y^{3} \cdot 2 - 8 (x \cdot x^{3}) y \cdot 2 - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

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

$y' = \frac{y (8 x - 40 x^{4} y^{3} \cdot 2 - 8 (x^{1} x^{3}) y \cdot 2 - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

$y' = \frac{y (8 x - 40 x^{4} y^{3} \cdot 2 - 8 x^{1 + 3} y \cdot 2 - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Add $1$ and $3$ .

$y' = \frac{y (8 x - 40 x^{4} y^{3} \cdot 2 - 8 x^{4} y \cdot 2 - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Simplify each term.

Multiply $2$ by $- 40$ .

$y' = \frac{y (8 x - 80 x^{4} y^{3} - 8 x^{4} y \cdot 2 - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Multiply $2$ by $- 8$ .

$y' = \frac{y (8 x - 80 x^{4} y^{3} - 16 x^{4} y - 5 y)}{2 x (80 x^{4} y^{3} + 5 y - 2 x)} - \frac{1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Combine the numerators over the common denominator.

$y' = \frac{y (8 x - 80 x^{4} y^{3} - 16 x^{4} y - 5 y) - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Simplify the numerator.

Apply the distributive property.

$y' = \frac{y (8 x) + y (- 80 x^{4} y^{3}) + y (- 16 x^{4} y) + y (- 5 y) - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Simplify.

Rewrite using the commutative property of multiplication.

$y' = \frac{8 y x + y (- 80 x^{4} y^{3}) + y (- 16 x^{4} y) + y (- 5 y) - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Rewrite using the commutative property of multiplication.

$y' = \frac{8 y x - 80 y (x^{4} y^{3}) + y (- 16 x^{4} y) + y (- 5 y) - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Rewrite using the commutative property of multiplication.

$y' = \frac{8 y x - 80 y (x^{4} y^{3}) - 16 y (x^{4} y) + y (- 5 y) - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Rewrite using the commutative property of multiplication.

$y' = \frac{8 y x - 80 y (x^{4} y^{3}) - 16 y (x^{4} y) - 5 y \cdot y - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Simplify each term.

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

Move $y^{3}$ .

$y' = \frac{8 y x - 80 (y^{3} y) x^{4} - 16 y (x^{4} y) - 5 y \cdot y - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

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

$y' = \frac{8 y x - 80 (y^{3} y^{1}) x^{4} - 16 y (x^{4} y) - 5 y \cdot y - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

$y' = \frac{8 y x - 80 y^{3 + 1} x^{4} - 16 y (x^{4} y) - 5 y \cdot y - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Add $3$ and $1$ .

$y' = \frac{8 y x - 80 y^{4} x^{4} - 16 y (x^{4} y) - 5 y \cdot y - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

Move $y$ .

$y' = \frac{8 y x - 80 y^{4} x^{4} - 16 (y \cdot y) x^{4} - 5 y \cdot y - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Multiply $y$ by $y$ .

$y' = \frac{8 y x - 80 y^{4} x^{4} - 16 y^{2} x^{4} - 5 y \cdot y - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

Move $y$ .

$y' = \frac{8 y x - 80 y^{4} x^{4} - 16 y^{2} x^{4} - 5 (y \cdot y) - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Multiply $y$ by $y$ .

$y' = \frac{8 y x - 80 y^{4} x^{4} - 16 y^{2} x^{4} - 5 y^{2} - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

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

$\frac{d y}{d x} = \frac{8 y x - 80 y^{4} x^{4} - 16 y^{2} x^{4} - 5 y^{2} - 1}{2 x (80 x^{4} y^{3} + 5 y - 2 x)}$

Find dy/dx arctan(4x^2y)=x+5xy^2

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