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(Susskind gives his own proof from 26:30 to 36:00 if you want to see his way.)

I am tempted to prove it from the chain rule, like this:

Let E(V, T) = F(V, S(V, T))

(How can we be sure there is such an F(V, S)?)

Then partially differentiating both sides with respect to V (using the chain rule on the right side) yields:

∂E(V, T)/∂V = ∂F(V, S)/∂V + ∂F(V, S)/∂S • ∂S(V, T)/∂V

Then the final step is to subtract the last term to the other side.

The arguments of each function makes clear what is held fixed.

Both E and F represent energy, but we cannot put F = E because they are different functions.

Is this a valid way to articulate/prove it? Is Susskind’s notation valid? Any clarification is appreciated.