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. 

First, don't be fooled by how derivatives are introduced in looking at slopes.  This is an artifact of how functions are introduced by graphing them.  A function is just a black box that takes in an input number and spits out an output number.  I cannot tell you how many people I've run into who think the derivative is the slope of a tangent line or that the integral is the area under a curve.  These are applications of derivatives and integrals, not fundamental definitions.
From a purely mathematical perspective, consider a Taylor series expansion:  in a neighborhood around a value $a$, the behavior of a function $f(x)$ could be well approximated by looking at the derivative:
$$f(x) \approx f(a) + (x-a) f'(a) + O([x-a]^2)$$
Such an expansion is at the heart of, for example, the small angle approximation: you can approximate the sine of an angle by the value of the angle itself, which simplifies problems in many applications, including optics.
The integral gives us a way to generalize weighted sums on a discrete number of points or objects to a continuously weighted sum on a continuous interval.  Consider, for instance, the notion of assigning probabilities to events and then performing a weighted sum to get the expected value.  The integral allows us to assign a probability density to an interval and to integrate that density, weighted by the value function, in an analogous way to the discrete case.
