What does it mean to prove statements or solve problems "from first principles"?

Question

When asked to solve or prove something from first principles, what do they mean by that? They expect you to use only axioms or basic properties or both or it can be a bit more subjective understanding of the term?

Answer 1

You have got the intuition right. First principle implies you use the fundamental definitions, properties or axioms for the problem at hand. For example, If you are asked to find the derivative of $\tan(x)$ from first principles, you would do something like this:

$ \frac{d}{dx}\big(\tan(x)\big)=\lim_{h\rightarrow 0} \bigg(\frac{\tan(x+h)-\tan(x)}{h}\bigg) $

You could use $\tan(x)=\frac{\sin(x)}{\cos(x)}$ and apply the quotient rule but that would not be considered first principles. You are not using basic definition but a rule/property that is a consequence of the main definition.

Similarly in integrating $f(x)=x^{2}$ doing it from first principles implies using the Riemann integration and showing that it converges and finding the value.

I hope that is clear.