Leibniz’s notation for calculus can be convenient but also misleading due to the introduction of irrelevant letters and confusion between functions and function values. The author argues that the proper way to think about notation is to distinguish between a function and its evaluation at a specific point. They emphasize the importance of being precise in notation and criticize the use of unnecessary letters, such as assigning meaning to the choice of variables. The chain rule is discussed in terms of precise notation and the relationship between functions and partial derivatives. The author then applies precise notation to a specific example, highlighting the distinction between the three functions involved and their respective derivatives.
https://math.stackexchange.com/questions/3266639/notation-for-partial-derivative-of-functions-of-functions