Im trying to find tangency points in a problem so I need to differentiate, $\displaystyle x^2+y^2=4000^2$

Why is the derivative of y^{2}=0? I thought by the laws of implicit differentiation that you just multiply the y term by its derivative which then follows the same rules as x does, so using the power rule would result with y^{2}=2y. Why does y^{2}=0 then when I plugin the formula into wolfram?