Mr. Integral
Math notes/Calculus III/Partial Derivatives and Local Change

Multivariable Chain Rules and Implicit Surfaces — Cheat sheet

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Core rule

ddtf(x(t),y(t),t)=fxx′+fyy′+ft,\frac d{dt}f(x(t),y(t),t)=f_xx'+f_yy'+f_t, zx=−Fx/Fz.z_x=-F_x/F_z.

Watch for

Include every dependency and evaluate at the composed point. Implicit differentiation as a local graph needs the solved-for partial to be nonzero.