# how to find the differential of an implicit function given as system of equations?

Given a implicit function tried to find the $$\frac{du}{dx}$$ and $$\frac{dv}{dx}$$ of $$u(x,y)$$ and $$v(x,y)$$ $$x + y + u + v = a, x^3 + y^3+u^3+v^3 =b$$ I differentiated these equations: $$dx +dy + du + dv = 0, 3x^2dx+3y^2dy+3u^2du ++3v^2dv = 0$$

$$\Rightarrow$$ $$du = - \frac{1}{u^2} \left (v^2+y^2+x^2 \right ) ;dv = - \frac{1}{v^2} \left (u^2+y^2+x^2 \right )$$ $$\Rightarrow$$ $$\frac{dv}{dx} = -\frac{x^2}{v^2};\frac{du}{dx} = -\frac{x^2}{u^2}$$

but actually its wrong and correct answer is $$\frac{du}{dx} = -\frac{v^2-x^2}{v^2-u^2}$$ and $$\frac{dv}{dx} = \frac{u^2-x^2}{v^2- u^2}$$ So, I want to know where am I wrong and how to find differentials correctly?

Since $$u$$ and $$v$$ are functions of $$x$$ and $$y$$, then instead of simply $$du$$, you would have $$\frac{du}{dx}dx+\frac{du}{dy}dy$$, and similarly for $$dv$$.
While this approach can get the answer, another approach would give it to you easier. First, do derivative with respect to $$x$$. At this point, you could arrange your results into a system of equations, with $$\frac{du}{dx}$$ and $$\frac{dv}{dx}$$ as variables. You would solve this system to get the values of those derivatives. Then, you can repeat the process for derivative with respect to $$y$$.