How do you differentiate #sqrt(xy) = x - 2y#?

To differentiate sqrt(xy) = x - 2y, we'll use implicit differentiation.

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

On the left side, the derivative of sqrt(xy) with respect to x is (1/2)(xy)^(-1/2)(y + x(dy/dx)).

On the right side, the derivative of x - 2y with respect to x is 1 - 2(dy/dx).

Now, equate the derivatives obtained from both sides and solve for dy/dx:

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

Now, solve for dy/dx:

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

Multiply both sides by (2sqrt(xy) + 2x):

dy/dx * (2sqrt(xy) + 2x) = y + x(dy/dx)

Expand and rearrange terms:

dy/dx * 2sqrt(xy) + dy/dx * 2x = y + x(dy/dx)

Subtract x(dy/dx) from both sides:

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

Factor out dy/dx:

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

Combine like terms:

dy/dx * (2sqrt(xy) + x) = y

Now, divide both sides by (2sqrt(xy) + x) to solve for dy/dx:

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