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. 

(in addition to OldCurmudgeon)
If a roller-coaster is too wild for you, consider a merry go round with constant rotation around $(0, 0)$.
You can notice by differentiating x(t) and y(t) twice that acceleration is always towards the rotation-axis and if you differentiate two more times, you'll see that the jounce-vector is a positive constant multiple of the position vector.
