# What is the implicit derivative of #y=(x-y)^2+4xy+5y^2 #?

The implicit derivative is

For simplicity, we can distribute the values in the parentheses and then implement quadratic multiplication.

We now simplify like terms to get:

There is your answer! Hope it helps!

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To find the implicit derivative of ( y = (x - y)^2 + 4xy + 5y^2 ), differentiate both sides of the equation with respect to ( x ).

[ \frac{d}{dx} [y] = \frac{d}{dx} [(x - y)^2 + 4xy + 5y^2] ]

Apply the chain rule and product rule where necessary.

[ \frac{dy}{dx} = \frac{d}{dx} [(x - y)^2] + \frac{d}{dx} [4xy] + \frac{d}{dx} [5y^2] ]

[ = 2(x - y) \frac{d}{dx} [x - y] + 4y + 4x\frac{dy}{dx} + 10y\frac{dy}{dx} ]

Now, differentiate ( x - y ) with respect to ( x ) to get ( \frac{d}{dx} [x - y] = 1 - \frac{dy}{dx} ).

[ \frac{dy}{dx} = 2(x - y) \left(1 - \frac{dy}{dx}\right) + 4y + 4x\frac{dy}{dx} + 10y\frac{dy}{dx} ]

Now, solve for ( \frac{dy}{dx} ) by isolating it on one side of the equation. This equation is implicit and can't be directly solved for ( \frac{dy}{dx} ).

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