Taking the initiative to go back and learn on your own what the schools failed to impart to you is time well spent. Respect.

@derekowens Well, it helps whenever you have passionate people that care about their teaching methods, and give that away for basically free. Kudos my man, you're amazing.

Hi, folks!

Yes!

​@@derekowens i wanted to thank you for helping me with a question i had 10 - 15 years ago. It was back when KZbin had an inbox. I asked you how to find a point of intersection given a point (x, y) and a line (x1, y1) and (x2, y2). When the point drops a "shadow" under 90 degrees on the line. Not sure if you understand my explanation. But i vaguely remember KZbin had an ability to show drawings. So you understood my poor English back then. Nothing special came out of that formula. i used it to make a silly game. But its the memory that i really value. Thank you for helping me back then xD

@@voteforhamsandwich1112 Wow, thanks for that encouraging note. It's very nice to be remembered, even after all this time, and I'm glad I was able to help some. Hope you're doing very well!

It is worth noting how rapidly these ideas took hold. The writings of Newton and Leibniz were adopted by the academic community almost in real time as they were developed. Newton was still alive and had not yet even moved to London when Bernoulli was writing about this rule, less than a decade after the publication of the Principia. And all of this was well over century before Cauchy began to formalize the theory of limits, and a full century and a half before Weierstrass. The point here is that a formal proof involving limits is actually unnecessary. History actually demonstrates this, as neither Newton nor Leibniz, nor anyone else at that time, used limits. Yet they *understood* these ideas and what these infinitesimal quantities represented. And if one understands this, then one understands when one can or cannot divide by dx. It was a conscious choice to approach the theorem this way, without the use of limits. The deliberate aim here is an *intuitive* proof, which I believe can make the concepts more clear and the understanding deeper. For another example, consider what Leibniz wrote to Wallace: "It is useful to consider quantities infinitely small such that when their ratio is sought, they may not be considered zero, but which are rejected as often as they occur with quantities incomparably greater. Thus if we have x + dx, then dx is rejected. Similarly we cannot have x dx and dx dx standing together as x dx is incomparably greater than dx dx. Hence if we are to differentiate uv, we write d(uv) = (u+dv)(v+dv) - uv = uv + vdu + udv + du dv - uv = v du + u dv" Where did the du dv go in this "proof"? If you understand the nature of these infinitesimal quantities, it makes complete sense. And then you can divide by dx on both sides and you have what we call the Product Rule.

Yes, it certainly does. You can think about the amount of light per square meter landing on earth and compare that the light per square meter landing on Mars, farther away. And on out to the more distant planets and beyond.