Find dx/dy x^2-2xy+y^2-6x+2y=0

$x^{2} - 2 x y + y^{2} - 6 x + 2 y = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (x^{2} - 2 x y + y^{2} - 6 x + 2 y) = \frac{d}{d y} (0)$

Step 2

Differentiate the left side of the equation.

By the Sum Rule, the derivative of $x^{2} - 2 x y + y^{2} - 6 x + 2 y$ with respect to $y$ is $\frac{d}{d y} [x^{2}] + \frac{d}{d y} [- 2 x y] + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 6 x] + \frac{d}{d y} [2 y]$ .

$\frac{d}{d y} [x^{2}] + \frac{d}{d y} [- 2 x y] + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 6 x] + \frac{d}{d y} [2 y]$

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

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

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

$\frac{d}{d u} [u^{2}] \frac{d}{d y} [x] + \frac{d}{d y} [- 2 x y] + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 6 x] + \frac{d}{d y} [2 y]$

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

$2 u \frac{d}{d y} [x] + \frac{d}{d y} [- 2 x y] + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 6 x] + \frac{d}{d y} [2 y]$

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

$2 x \frac{d}{d y} [x] + \frac{d}{d y} [- 2 x y] + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 6 x] + \frac{d}{d y} [2 y]$

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

$2 x x' + \frac{d}{d y} [- 2 x y] + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 6 x] + \frac{d}{d y} [2 y]$

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

$2 x x' - 2 \frac{d}{d y} [x y] + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 6 x] + \frac{d}{d y} [2 y]$

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 $g (y) = y$ .

$2 x x' - 2 (x \frac{d}{d y} [y] + y \frac{d}{d y} [x]) + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 6 x] + \frac{d}{d y} [2 y]$

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

$2 x x' - 2 (x \cdot 1 + y \frac{d}{d y} [x]) + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 6 x] + \frac{d}{d y} [2 y]$

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

$2 x x' - 2 (x \cdot 1 + y x') + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 6 x] + \frac{d}{d y} [2 y]$

Multiply $x$ by $1$ .

$2 x x' - 2 (x + y x') + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 6 x] + \frac{d}{d y} [2 y]$

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

$2 x x' - 2 (x + y x') + 2 y + \frac{d}{d y} [- 6 x] + \frac{d}{d y} [2 y]$

Evaluate $\frac{d}{d y} [- 6 x]$ .

Since $- 6$ is constant with respect to $y$ , the derivative of $- 6 x$ with respect to $y$ is $- 6 \frac{d}{d y} [x]$ .

$2 x x' - 2 (x + y x') + 2 y - 6 \frac{d}{d y} [x] + \frac{d}{d y} [2 y]$

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

$2 x x' - 2 (x + y x') + 2 y - 6 x' + \frac{d}{d y} [2 y]$

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

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

$2 x x' - 2 (x + y x') + 2 y - 6 x' + 2 \frac{d}{d y} [y]$

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

$2 x x' - 2 (x + y x') + 2 y - 6 x' + 2 \cdot 1$

Multiply $2$ by $1$ .

$2 x x' - 2 (x + y x') + 2 y - 6 x' + 2$

Simplify.

Apply the distributive property.

$2 x x' - 2 x - 2 (y x') + 2 y - 6 x' + 2$

Remove unnecessary parentheses.

$2 x x' - 2 x - 2 y x' + 2 y - 6 x' + 2$

Reorder terms.

$2 x x' - 2 y x' - 2 x + 2 y + 2 - 6 x'$

Step 3

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

$0$

Step 4

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

$2 x x' - 2 y x' - 2 x + 2 y + 2 - 6 x' = 0$

Step 5

Solve for $x'$ .

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

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

$2 x x' - 2 y x' + 2 y + 2 - 6 x' = 2 x$

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

$2 x x' - 2 y x' + 2 - 6 x' = 2 x - 2 y$

Subtract $2$ from both sides of the equation.

$2 x x' - 2 y x' - 6 x' = 2 x - 2 y - 2$

Factor $2 x'$ out of $2 x x' - 2 y x' - 6 x'$ .

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

$2 x' x - 2 y x' - 6 x' = 2 x - 2 y - 2$

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

$2 x' x + 2 x' (- y) - 6 x' = 2 x - 2 y - 2$

Factor $2 x'$ out of $- 6 x'$ .

$2 x' x + 2 x' (- y) + 2 x' \cdot - 3 = 2 x - 2 y - 2$

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

$2 x' (x - y) + 2 x' \cdot - 3 = 2 x - 2 y - 2$

Factor $2 x'$ out of $2 x' (x - y) + 2 x' \cdot - 3$ .

$2 x' (x - y - 3) = 2 x - 2 y - 2$

Divide each term in $2 x' (x - y - 3) = 2 x - 2 y - 2$ by $2 (x - y - 3)$ and simplify.

Divide each term in $2 x' (x - y - 3) = 2 x - 2 y - 2$ by $2 (x - y - 3)$ .

$\frac{2 x' (x - y - 3)}{2 (x - y - 3)} = \frac{2 x}{2 (x - y - 3)} + \frac{- 2 y}{2 (x - y - 3)} + \frac{- 2}{2 (x - y - 3)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x' (x - y - 3)}{2 (x - y - 3)} = \frac{2 x}{2 (x - y - 3)} + \frac{- 2 y}{2 (x - y - 3)} + \frac{- 2}{2 (x - y - 3)}$

Rewrite the expression.

$\frac{x' (x - y - 3)}{x - y - 3} = \frac{2 x}{2 (x - y - 3)} + \frac{- 2 y}{2 (x - y - 3)} + \frac{- 2}{2 (x - y - 3)}$

Cancel the common factor of $x - y - 3$ .

Cancel the common factor.

$\frac{x' (x - y - 3)}{x - y - 3} = \frac{2 x}{2 (x - y - 3)} + \frac{- 2 y}{2 (x - y - 3)} + \frac{- 2}{2 (x - y - 3)}$

Divide $x'$ by $1$ .

$x' = \frac{2 x}{2 (x - y - 3)} + \frac{- 2 y}{2 (x - y - 3)} + \frac{- 2}{2 (x - y - 3)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

$x' = \frac{2 x}{2 (x - y - 3)} + \frac{- 2 y}{2 (x - y - 3)} + \frac{- 2}{2 (x - y - 3)}$

Rewrite the expression.

$x' = \frac{x}{x - y - 3} + \frac{- 2 y}{2 (x - y - 3)} + \frac{- 2}{2 (x - y - 3)}$

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

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

$x' = \frac{x}{x - y - 3} + \frac{2 (- y)}{2 (x - y - 3)} + \frac{- 2}{2 (x - y - 3)}$

Cancel the common factors.

Cancel the common factor.

$x' = \frac{x}{x - y - 3} + \frac{2 (- y)}{2 (x - y - 3)} + \frac{- 2}{2 (x - y - 3)}$

Rewrite the expression.

$x' = \frac{x}{x - y - 3} + \frac{- y}{x - y - 3} + \frac{- 2}{2 (x - y - 3)}$

Move the negative in front of the fraction.

$x' = \frac{x}{x - y - 3} - \frac{y}{x - y - 3} + \frac{- 2}{2 (x - y - 3)}$

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

Factor $2$ out of $- 2$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

$x' = \frac{x}{x - y - 3} - \frac{y}{x - y - 3} + \frac{- 1}{x - y - 3}$

Move the negative in front of the fraction.

$x' = \frac{x}{x - y - 3} - \frac{y}{x - y - 3} - \frac{1}{x - y - 3}$

Combine into one fraction.

Combine the numerators over the common denominator.

$x' = \frac{x - y}{x - y - 3} - \frac{1}{x - y - 3}$

Combine the numerators over the common denominator.

$x' = \frac{x - y - 1}{x - y - 3}$

Step 6

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

$\frac{d x}{d y} = \frac{x - y - 1}{x - y - 3}$

Find dx/dy x^2-2xy+y^2-6x+2y=0

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