3 ms·

Funnily, I disagree with almost every point in the most upvoted answer. For me (as a physicist by training), the f' notation is a shorthand, and df/dx is the mo

Funnily, I disagree with almost every point in the most upvoted answer. For me (as a physicist by training), the f' notation is a shorthand, and df/dx is the more clear notation. Because especially in physics, you often deal with functions that are dependent on multiple variables. Do I want to differentiate wrt. time or position? You can use "f dot" for the time derivative, but in more complex cases you are out of luck. How can you distinguish between f(x) and f(t) which are very different functions? And 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. 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. I don't recall seeing an explicit proof why it works (except for some specific cases), or a list of exact rules, but I'm sure that exists and it would have been neat to have seen that in my studies. [1] calculus here in the sense of German Kalkül, a notational system and a method of mechanically manipulating symbols, and not neccessarily meaning "differential and integral calculation", although "the" calculus is the prime example of "a calculus" of course.

pinkmuffinere 3y ago
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

pinkmuffinere 3y ago

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