What is the significance of the slope of the tangent line of a function? Why is the derivative so important? As I finished calc 1. I can use the product rule and chain rule and resolve integrals. But I feel like its too mechanical for my taste. I know the procedure and I execute on paper without really understanding or experiencing the "ahaa moment". 

For example, when I was learning geometry in elementary school, the "ahaa moment" for me was when I had to move furniture in my room and needed to find areas of stuff. I'm trying to find the equivalent application of the derivative and integral as I learn calculus. Could someone demystify this? 

Thank you in advance.
A slope is a rate of change.  If a car moves at 30 miles per hour, and you graph its distance traveled as a function of time, the slope of that graph is 30 miles per hour.

Isaac Newton was the first person ever to show that the laws of nature that govern objects we see every day here on Earth are the same as the laws of nature that govern things in the heavens: planets, moons, comets (including the planet Earth as a whole).

In order to do that he had to think about the rates at which they move as a function of their locations.

Rates of change also occur in the following way: Newton asked himself whether the gravitational pull of the earth on the moon or other orbiting object is the same as if the whole mass of the earth were concentrated at the center.  (The bottom line, after much effort, turns out to be "yes".)  So how do you find the sum of the infinitely many infinitely small quantities involved?  The answer is that you ask how fast that sum changes as progressively more of the earth is taken into account.  This is not a rate of physical motion as time passes, but is something else.

Unfortunately far too much of the conventional mathematics curriculum is designed to prepare students for later courses, in such a way that you find out the motivations only if you take certain later courses that most of the students will never take.  It leaves students not understanding the whole thing.  Honest education does not do that.  A few professors here and there are working on calculus courses that are in that respect honest.  At least two professors at Macalester College have been working on that, and at various other places that I can't name right now.
An application, which if applied world wide on all roads, would make it impossible to go over the speed limit and go unnoticed.

If you place sensors with timers that correctly identify passing cars throughout every stretch of road spaced 1km apart and the worldwide speed limit is now 50 km/h = 50 km/3600 seconds, a car going at top legal speed (50km/h) would take 1/50th of an hour = 3600seconds/50 = 72 seconds to cross adjacent sensors.
If a car takes less than 72 seconds to cross adjacent sensors, then the car MUST have gone over the speed limit at some point between the sensors. This conclusion comes from the fact that we have a mapping between time and distance, which allows us to make accurate (in my example not exact) deductions about the collected information.

If it still isn't clear enough, I will demonstrate how exactly the program coordinating the sensors would go:

var max_speed=50
Sensor A:Car[x] Passes at Time Stamp 1:15:15PM
Sensor B:Car[x] Passes at Time Stamp 1:16:10PM
//average* speed=difference in distance/difference in time
//[f(x2)-f(x1)]/[x2-x1]=1km/55seconds
//1km/55seconds][60seconds/1min][60min/1h]=[3600/55]km/h=65.45km/h
var Car[x].speed=[1/Date(1:16:10)-Date(1:15:15)][3600]=65.45/h
65km/h&gt;50km/h = EMIT SPEEEDING TICKET FOR Car[x]


*average speed: you might actually go at various instantaneous speeds throughout continuous length of road (even if you put it on cruise control).