If we have three implicit functions as following: $$f(x,y,z)=0$$ $$g(x,t)=0$$ $$M(x,y)=0$$ First, if I want to compute the derivative of z w.r.t x , will it be as following? $$\frac{dz}{dx}=z_x+z_y\frac{dy}{dx}$$ where $$\frac{dy}{dx}=-\frac{M_x}{M_y}$$ $$z_x=-\frac{f_x}{f_z}$$ $$z_y=-\frac{f_y}{f_z}$$ I think this is wrong because we can solve it using this formula only if z is an explicit function of x and y, but here it is an implicit function... So how can we compute the derivative of z w.r.t. x if this formula should not be applied here?
Second, if I want to compute the derivative of z w.r.t. t : my trial is as following $$\frac{dz}{dt}=(z_x+z_y\frac{dy}{dx})\frac{dx}{dt}$$ where $$\frac{dx}{dt}=-\frac{g_t}{g_x}$$ But also I think this wrong solution for the same reason.. So how can we solve this if my answer is wrong?