Find dx/dy 2xy-y^2=1

$2 x y - y^{2} = 1$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (2 x y - y^{2}) = \frac{d}{d y} (1)$

Step 2

Differentiate the left side of the equation.

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

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

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

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

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

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = y$ .

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

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

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

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

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

Multiply $x$ by $1$ .

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

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

Since $- 1$ is constant with respect to $y$ , the derivative of $- y^{2}$ with respect to $y$ is $- \frac{d}{d y} [y^{2}]$ .

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

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

$2 (x + y x') - (2 y)$

Multiply $2$ by $- 1$ .

$2 (x + y x') - 2 y$

Simplify.

Apply the distributive property.

$2 x + 2 (y x') - 2 y$

Reorder terms.

$2 y x' + 2 x - 2 y$

Step 3

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$0$

Step 4

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

$2 y x' + 2 x - 2 y = 0$

Step 5

Solve for $x'$ .

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

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

$2 y x' - 2 y = - 2 x$

Add $2 y$ to both sides of the equation.

$2 y x' = - 2 x + 2 y$

Divide each term in $2 y x' = - 2 x + 2 y$ by $2 y$ and simplify.

Divide each term in $2 y x' = - 2 x + 2 y$ by $2 y$ .

$\frac{2 y x'}{2 y} = \frac{- 2 x}{2 y} + \frac{2 y}{2 y}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y x'}{2 y} = \frac{- 2 x}{2 y} + \frac{2 y}{2 y}$

Rewrite the expression.

$\frac{y x'}{y} = \frac{- 2 x}{2 y} + \frac{2 y}{2 y}$

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{y x'}{y} = \frac{- 2 x}{2 y} + \frac{2 y}{2 y}$

Divide $x'$ by $1$ .

$x' = \frac{- 2 x}{2 y} + \frac{2 y}{2 y}$

Simplify the right side.

Simplify each term.

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

Factor $2$ out of $- 2 x$ .

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

Cancel the common factors.

Factor $2$ out of $2 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

$x' = \frac{- x}{y} + \frac{2 y}{2 y}$

Move the negative in front of the fraction.

$x' = - \frac{x}{y} + \frac{2 y}{2 y}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x' = - \frac{x}{y} + \frac{2 y}{2 y}$

Rewrite the expression.

$x' = - \frac{x}{y} + \frac{y}{y}$

Cancel the common factor of $y$ .

Cancel the common factor.

$x' = - \frac{x}{y} + \frac{y}{y}$

Rewrite the expression.

$x' = - \frac{x}{y} + 1$

Step 6

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

$\frac{d x}{d y} = - \frac{x}{y} + 1$

Find dx/dy 2xy-y^2=1

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