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. 

The differential is the slope or the rate of change.
On a roller-coaster:


*

*Your velocity is the rate of change of position.

*Your acceleration is the rate of change of your velocity.

*Your jerk is the rate of change of your acceleration.

*Your jounce is the rate of change of your jerk.

