A question about convention of writing identities and formulas I'm only a third year undergraduate math student, but I've seen a decent amount of formulas, identities, and equations (at least enough to formulate such a question). However, I haven't actually thought about writing convention until now. Why is it that certain terms are universally written first? For example, I almost always see the definition of a derivative written as
$$\lim_{h \to 0}\frac{f(x+h) - f(x)}{h}$$
instead of
$$\lim_{h \to 0}\frac{-f(x) + f(x+h)}{h}$$
I tutor Calculus 1, and one thing I notice is that students sometimes forget to distribute the $-1$ when $f(x)$ has multiple terms. However, when I write it like the latter, it seems to become more obvious to some students that parentheses are very important. I would think that the former is conventional simply because it saves the time of writing an extra plus sign.
A more relevant example that better corresponds to my question is the probability of events for a Poisson distribution, which is given by
$$f(k) = \frac{\lambda^ke^{-\lambda}}{k!}$$
Why is it conventional to write $e^{-\lambda}$ instead of simply putting it in the denominator?
$$f(k) = \frac{\lambda^k}{k!\,e^\lambda}$$
It seems odd to me because of what I learned in my own Calculus 1 course that continues to stick with me today. When we take the derivative of, for example, $\sqrt{x+4}$, we were taught to first write it as $(x+4)^{\frac{1}{2}}$ and then use the power rule and chain rule to obtain $\frac{1}{2}(x+4)^{-\frac{1}{2}}$, but points were marked off unless we simplified it to $\frac{1}{2\sqrt{x+4}}$. I can see why writing it this way would be advised because it's easier to see what the domain is, but on an exam which doesn't require domain specification, it may save a bit of time to leave it as a negative exponent.
A final example is one of Ptolemy's trigonometric identities: $\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)$. Why is it traditional to write the terms in that order?
 A: Most of your questions can be answered by either "because it's easier", "because it's clearer" or "because we've always written it like that"
Let's go step by step:


*

*Why do we write $f(x+h)-f(x)$ instead of the other way around? Two reasons: one, as you noticed, it's simple. But two is more important: It's clearer. If I ask you "what's the difference between $a$ and $b$", the answer will be $a-b$, not $-b+a$. The point of the derivative is that the enumerator has the difference between $f(x+h)$ and $f(x)$.

*For the Poisson distribution, writing $e^{-\lambda}$ on top instead of $e^\lambda$ at the bottom is, first of all convention. It's been written like that for a long time. However, if you ask me, it's also clearer. The expression $e^{-\lambda}$ isn't the same as the expression $x^{-\frac12}$. Often, as a mathematitian, I consider $e^{-\lambda}$ as a entity "on it's own". Let me explain that: If I see the expression $x^{-\frac12}$, I think: (a) OK, I have a minus sign on top. (b) So this thing, whatever it is, is one divided by something simpler. (c) And the "simpler" thing is $x^\frac12$, which is $\sqrt x$. (d) So the whole thing is the inverse of the square of $x$. On the other hand, If I see $e^{-\lambda}$, I don't go through hoops. I just thing "oh, that's the "suppresion" thing that goes to $0$ very very quickly.

*Ptolemy's identity is written like that because it's always been written like that. You have to pick one or the other, and it's then easier if you keep sticking to it once you picked it. Easier on the memory.

