TITLE: Finding the derivative of function with domain empty set
QUESTION [20 upvotes]: I was asked to find the derivative of the function
$$
\sin^{-1}(e^x + 3).
$$
First I was thinking of the chain rule, but something looked strange about the function. The domain of $\sin^{-1}$ is $[-1,1]$ and $e^x + 3$ is always strictly greater than $3$. I think that there is a mistake in the problem, but assuming that there isn't: what exactly is the derivative of a function like this one that "doesn't exist"? (Is it true to say that the function doesn't exist?)

REPLY [4 votes]: The fundamental part of definition of a derivative is that the function must be defined at the point under consideration. The defining equation $$f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$$ does not make sense unless $f(x)$ makes sense (plus you must also ensure that $f(x + h)$ makes sense for sufficiently small values of $h$). So the correct answer to the problem is that since the function in question does not exist there is no point in even considering the derivative of the function.
Some answers mention the fact that the function could be thought of a complex function of a real variable. This is nowhere specifically mentioned in the problem and if we really want to be imaginative enough, we may well consider it to be a function of a complex variable. And this changes the whole nature of the problem. Functions of a complex variable are altogether a very different beast from the usual functions of a real variable which are seen in introductory calculus courses.
Another aspect which is mentioned in other answers is about formal differentiation. This aspect is only for polynomials (and also power series as user Ethan Bolker mentions in comment) because for such functions it can be developed formally without any reference to limit definition (and the laws of derivatives remain same). The approach does not work for arbitrary functions. One may try to develop a formal approach restricting oneself to elementary functions, but then the rules of differentiation must be taken for granted as axioms and we lose theorems of differential calculus so the whole idea of formal differentiation does not sound very appealing. Unfortunately the idea that derivatives can be handled formally (without appeal to limit processes) is something which has caught on and routinely used by students in evaluation of limits via L'Hospital's Rule.

It is best to treat this problem as a typo/mistake and luckily this one is easy to spot. A slightly "difficult to spot" mistake occurred long back in an actual competitive exam in India.