Named arguments are great, but mathematical notation doesn’t have them. x(t) is not a named argument to f(x(t), t), so notation like ∂f(x(t), t)/∂x(t) is nonse

Named arguments are great, but mathematical notation doesn’t have them. x(t) is not a named argument to f(x(t), t), so notation like ∂f(x(t), t)/∂x(t) is nonsense. If we did have real named arguments like f(a=x(t), b=t), where a and b are fixed by the definition of f rather than arbitrary names to be made up on each invocation, then maybe it would make sense to write something like (∂f/∂a)(a=x(t), b=t). Though that’s still pretty far from Leibniz notation, where ∂f/∂a is somehow supposed to be a numerical value depending on a, not a function to which arguments must be supplied. But what I think is really lacking in mathematical notation is explicit lambda abstraction: we should be able to write (λt. f(x(t), t))'(t) = (λa. f(a, t))'(x(t)) x'(t) + (λb. f(x(t), b))'(t), which reuses the ordinary one-variable derivative and has none of these ambiguities.