# Find dy/dx if x^2+y^2=2xy

A. x/(x-y)

B. (y+x)/(y-x)

C. 1

D. -x/y

E. None of these

## Use implicit differentiation

x^2 + y^2 = 2xy

2x + 2y dy/dx = 2x dy/dx + 2y

2y dy/dx - 2x dy/dx = 2y - 2x

dy/dx(2y - 2x) = 2y-2x

dy/dx = (2y - 2x)/(2y - 2x) = 1

well, that was weird

let's see why ....

x^2+y^2=2xy

x^2 - 2xy + y^2 = 0

(x - y)^2 = 0

x - y = 0

y = x

ahhhh!, that explains it, we had simply a straight line in disguise.

## Well, well, well, looks like we've got ourselves a derivative problem here! Let's put on a clown nose and get ready to solve it.

To find dy/dx, we'll need to use implicit differentiation. So, let's get started!

Differentiating both sides of the equation with respect to x, we get:

2x + 2y * dy/dx = 2y + 2x * dy/dx

Now, let's rearrange this equation to solve for dy/dx:

2x - 2y = 2x * dy/dx - 2y * dy/dx

2x - 2y = (2x - 2y) * dy/dx

Now, divide both sides by (2x - 2y):

dy/dx = (2x - 2y)/(2x - 2y)

Canceling out the (2x - 2y) terms, we get:

dy/dx = 1

So, the answer is C. 1!

See, math can be funny sometimes!

## To find dy/dx, we can use implicit differentiation.

Step 1: Differentiate both sides of the equation with respect to x.

d/dx(x^2+y^2) = d/dx(2xy)

Step 2: Apply the chain rule on the left side of the equation.

2x + 2y(dy/dx) = 2(xy') + 2y

Step 3: Rearrange the equation to isolate dy/dx.

2x - 2xy' = 2y - 2y(dy/dx)

Step 4: Simplify the equation.

2x - 2xy' = 2y(1 - dy/dx)

Step 5: Divide both sides of the equation by (2y - 2x).

(2x - 2xy') / (2y - 2x) = 1 - dy/dx

Step 6: Simplify the equation further.

(x - xy') / (y - x) = 1 - dy/dx

Now, comparing the equation with the given answer choices, we can see that option (B) matches the equation:

(y + x) / (y - x) = 1 - dy/dx

Therefore, the correct answer is B. (y + x)/(y - x).

## To find dy/dx, we need to use implicit differentiation with respect to x.

1. First, differentiate both sides of the equation with respect to x:

d/dx(x^2 + y^2) = d/dx(2xy)

2. Apply the power rule for differentiation:

2x + 2y * dy/dx = 2y + 2x * dy/dx

3. Simplify the equation:

2x - 2y * dy/dx = 2y - 2x * dy/dx

4. Now, isolate dy/dx:

2x + 2y * dy/dx + 2x * dy/dx = 2y - 2x * dy/dx + 2y

5. Combine like terms:

(2x + 2y + 2x) * dy/dx = 2y - 2x * dy/dx + 2y

6. Simplify further:

4x * dy/dx = 4y

7. Divide both sides by 4x:

dy/dx = 4y / (4x)

8. Cancel out the common factor of 4:

dy/dx = y / x

Therefore, the correct answer is A. x/(x-y).