# Solving/Proving from first principles?

When asked to solve/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?

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Specifically, which area of math are you talking about? Calculus? – Calvin Lin Dec 30 '12 at 3:52
It's normally axioms and definitions. Calculating a derivative from first principles, for example, would involve doing it directly from the definition of the derivative rather than the chain rule, product rule, etc. – Robert Mastragostino Dec 30 '12 at 3:53
That is my main question. Sometimes I think it depends on the area, but if they ask for "first principles" I would expect to have an unambigous meaning of that term. – aortizmena Dec 30 '12 at 3:56
"First principles" is always local. – André Nicolas Dec 30 '12 at 5:13

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.