TITLE: Why do differentiation rules work? What's the intuition behind them? (Not asking for proofs)
QUESTION [41 upvotes]: Differentiation rules have been bugging me ever since I took Basic Calculus. I thought I'd develop some intuitive understanding of them eventually, but so far all my other math courses (including Multivariable Calculus) take the rules for granted.
I know how to prove some of the rules. The problem is that algebra manipulation alone isn't quite convincing to me. Is there any possibility of understanding why the algebra happens to work that way? For example, why do the slopes of the tangent line to the parabola x^2 happen to be determined by 2x? Looking at it graphically, there's no way I could've told that.
Any sources covering this issue (books; internet sites; etc) would be very greatly appreciated. Thanks in advance.

REPLY [2 votes]: If you look for intuition, I cannot recommend the series "Essence of Calculus" by 3Blue1Brown enough!