You're totally right... but the previous poster is too. I encounter many students who really freak out at f(t) = y vs f(t) = x vs f(x) = y. Obviously they don't understand very well. I go back and forth between using notation as a crutch (always stay consistent so we can focus on the topic instead of the notation) and trying to really wean students from notation-dependence (let's do f(smily face) = y today! let's rotate all the axes and draw this with the z-axis sticking out!).
When people have very fragile understanding, they may be genuinely unsure if derivative with respect to x is computed the same way as derivative with respect to n. After all, x is always a real-valued variable and n is always a non-negative integer.... right?
This is an enormously serious problem with humans. All humans. Cognitive biases, miscommunication, misinterpreting patterns and data and evidence, inability to differentiate what something looks like from what's inside, inability to believe statistics that contradict personal experience, the list goes on and on...
People who have learned math and are comfortable with it's symbols and abstractions are less susceptible, but all people are susceptible to this confusion.
I'm certain you would be confused by non-standard usage of notation. It's not just symbols and that any symbol can be used. Try reading or speaking English with the adjectives in an order that is not expected. It's confusing as hell but grammatically correct. Our brains get trained to see things in a certain way and when the symbols get jumbled up so do we.
(x1(x2+x3) - x1(x2))/x3
is simply not as understandable to a first year calculus student as
(f(x+h) - f(x))/h
I've been teaching university level mathematics for 25 years and I know mixing up symbols confuses people. Try reading Newton's Principia. It's really hard to know what he's talking about.
EDIT: Try this equation. What famous one does it represent?
Here's something else to consider. There used to be obfuscated C programming contests. If symbols are just symbols and using them in nonstandard ways isn't confusing then shouldn't this mean that there is no such thing as obfuscated C programs? Does the existence of obfuscated C programming contests imply that programmers have a serious problem with understanding vs rote learning?
When people have very fragile understanding, they may be genuinely unsure if derivative with respect to x is computed the same way as derivative with respect to n. After all, x is always a real-valued variable and n is always a non-negative integer.... right?