Many of your objections are addressed in the stackexchange post: 1. "You often deal with functions that are dependent on multiple variables. Do I want to diffe

Many of your objections are addressed in the stackexchange post: 1. "You often deal with functions that are dependent on multiple variables. Do I want to differentiate wrt. time or position?" -- the proposed solution is to put a subscript under the function name, to clarify whether you're differentiating wrt time, position, etc 2. "How can you distinguish between f(x) and f(t) which are very different functions?" -- the answer claims (and I agree) that using f(x) and f(t) to represent different functions is bad notation. If x and t are variables, surely f(some_variable) == f(other_variable). If x and t are specified values, then f(value1) may not equal f(value2), but f still _really really_ looks like the same function, just evaluated at different points. Better is to use different function names, like 'f' and 'g'. 3. "what if the variables depend on each other? You need a notation that distinguishes between treating "x" as a variable, and "x" as something that is dependent on some other variable" -- there is a proposed way to represent composition of functions. I don't want to dive into latex editing on hn, but you can see it in the post 4. "The nice thing about Leibnitz calculus [1] is that you can do things like reduce fractions with dx'es, you can flip things around and calculate dx/df, the chain rule is not a rule that you have to memorize but just an obvious expansion etc., and it mostly just works" -- it does indeed work sometimes, but this is not rigorous. When it works, it does so because you're using it in a domain where it just happens to work. "Cancelling" dx's is not reliable, and will sometimes lead to error. I'll admit I find the chain rule mnemonic convenient though