If you have a function [tex] x = x(u,t) [/tex] then does [itex]u[/itex] necessarily depend on [itex]x[/itex] and [itex]t[/itex]? so [itex]u = (x,t)[/itex] For example, if [itex] x(u,t)=u^2 t[/itex] it seems that because [itex] t=x/u^2 [/itex], [itex] t=t(x,u) [/itex] I am having difficulty working out the general equation for [itex] dz \over dx [/itex] if [tex]z=z(x,y,t) [/tex] [tex]x=x(u,t) [/tex][tex]y=y(u,v,t) [/tex] The chain rule suggests it should be: But I don't know how to simplify [itex] dy \over dx[/itex] or [itex] dt \over dx[/itex], as neither [itex]y[/itex] nor [itex]t[/itex] depends on [itex]x[/itex] directly or indirectly. I can easily work out [itex]dz \over dt[/itex] because [itex]t[/itex] is at the bottom of the tree of dependency, but [itex]x[/itex] is not. Can you have cyclic dependencies? Or are all dependencies cyclic? What does this mean for the chain rule? Thanks.