Proof of the Multivariable Chain Rule

[chetanraina]

At 4:25 how is equation *1 equal to df/dy*dy/dt ?

[LaureanoLuna]

For the mean value theorem to hold, the function must be differentiable at all points between t and t+h, not just continuous (this is no real problem for your proof, though). However, I see no point in using that theorem, for at the limit yo will equate c_x and c_y with t. In fact, the core of your proof is equating star_2 with f_x·dx/dt, which is doubtful: one should take the derivative of f at a point x(t), y(t), for some t, but I'm not sure you can use the vanishing quantity h to define the point x(t), y(t+h) in order to take the derivative of f at that point. In fact, you seem to be passing to the limit twice: first, to turn mean changes into derivatives; then, to equate c_x and c_y to t. I'm no expert, I may be wrong but your proof is unconvincing to me.