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My understanding of partial $\left( \frac{\partial}{\partial} \right)$ and total $\left( \frac{d}{d} \right)$ differentiation/derivative is that assuming $f(x_1, x_2, ...,x_n )$ where $x_i$s are not necessarily independent:

$$\frac{d f}{ dx_i}=\sum^n_1 \left(\frac{\partial f}{\partial x_j}\frac{d x_j}{dx_i} \right)$$

Where $\frac{\partial f}{\partial x_i}$ is the symbolic derivative of the equation $f(x_1, x_2, ...,x_n )$ assuming all $x_j$s except $x_i$ are constants. Of course when $x_i$s are independent:

$$\frac{\partial f}{\partial x_i}=\frac{d f}{ dx_i}$$

But in thermodynamics I see that they have this exact differential

$$\left(\frac{\partial f}{\partial x_i} \right)_{x_j}$$

which to me looks exactly the same as partial differential. For example see these videos of thermodynamic lectures from MIT. I find this concept/notation redundant and confusing. I would appreciate if you could explain the difference between partial and exact differentials and give me a tangible example when they are not the same.

P.S.1. This post also approves my point:

In fact, the constancy of the other variables is implicit in the partial differential notation (∂/∂x) but it is customary to write the variables that are constant under the derivative when discussing thermodynamics, just to keep track of what other variables we were considering in that particular case.

Which if true, is an awful idea. Partial differential equations are already long and confusing enough without these redundant notations. Why on earth should we make it even more difficult?