This is from my text, "Linear Algebra" by Serge Lang, pg 11: -The two functions et, e2t are linearly independent. To prove this, suppose that there are numbers a, b such that: aet + be2t=0 (for all values of t). Differentiate this relation. We obtain aet + 2be2t = 0. Subtract the first from the second relation. We obtain be2t=0, and hence b=0. From teh first relation, it follows that aet=0, and hence a=0. Hence et, e2t are linearly independent. I'm confused as to the "Differentiate this relation". I see it creates a system of equations which can then be used to solve for linear independence, but why does it work?