# How do you find the derivative of #2xy=y^2#?

Differentiating implicitly gets us

Further discussion

graph{y(y-2x)=0 [-17.19, 14.86, -9.43, 6.59]}

Rewriting the derivative

Checking the three cases, we see that

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To find the derivative of (2xy = y^2), you can use implicit differentiation. Start by differentiating both sides of the equation with respect to (x).

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

Apply the product rule to the left side, and the chain rule to the right side:

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

Now, solve for (\frac{dy}{dx}):

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

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

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

(\frac{dy}{dx} = \frac{-2y}{2x - 2y})

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