What does it mean to prove statements or solve problems "from first principles"? 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? 
 A: 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.
A: First principals are basic, evident assumptions that we begin with whenever we seek to start proving stuff. Their fairly similar to axioms, but I like to think as axioms as being a minimal set of assumptions, whereas your first principals include all other obvious facts too.
Proving something "from first principles" contrasts with, what I've heard some people refer to as, driving in a nail with a sledgehammer. It can certainly make your problem easy to solve, but sometimes it doesn't feel very elegant, especially if your problem shouldn't be too tough to solve. So sometimes instructors will ask that you prove a statement "from first principles" to see that you can come up with the elegant argument. A common example of this is the use of L'Hôpital's rule to evaluate certain limits. L'Hôpital's is a sledgehammer: it's not immediately obvious why L'Hôpital's rule works, and its proof isn't straightforward. Using L'Hôpital's rule to analyze an otherwise easy-to-evaluate limit seems inappropriate.
