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Any insight would be greatly appreciated!

What I do understand:

- It’s kind of like a multivariable chain rule
- The partials terms tell us how the function changes with respect to a parameter and the full derivatives tell us how the parameters themselves change
- The result is a scalar function that represents how much the original function changes at every point

What I don't understand:

- If it is a multivariable chain rule then why is it written with partials and not as $$\frac{df(u,v)}{dx}= \frac{\mathrm{d}f }{\mathrm{d} u}\frac{\mathrm{d}u }{\mathrm{d} x}+\frac{\mathrm{d}f }{\mathrm{d} v}\frac{\mathrm{d}v }{\mathrm{d} x}$$
- It seems there is no necessary reason for terms to be summed instead of combined in some other functional form (multiplied for example)
- What are the conditions that allow us to multiply by the denominator and arrive at the total differential? $$df = \frac{\partial f}{\partial u}du+\frac{\partial f}{\partial v}dv$$Is it really ok to be so cavalier manipulating differentials and derivatives?
- Is there a connection between the total derivative and functional derivative that makes the expressions look so similar, or is it just a cosmetic similarity?