What's the purpose of finding derivatives? I don't understand why you would need to find a derivative, which is the tangent of the slope of a point on a graph, to find something like "How far did Bob fall". Confusing. Please help. Thank you!

Answer 1

If you had a graph of distance versus time for example you might want to know the speed at some specific time. That is the slope at that time
s = dx/dt
again the magnitude of acceleration is the time derivative of speed
|a| = ds/dt = d/dt(dx/dt) = d^2x/dt^2
note that I said distance and speed and magnitude of acceleration. Position , velocity, and acceleration also have direction so they are vectors with both magnitude and direction. To measure speed you need a speedometer. To measure velocity you need a speedometer and a compass.
Of course if you want to do physics or hydrodynamics or design bridges you need much more.
Now as for Bob falling
acceleration = -g = d^2h/dt^2
velocity = dh/dt = -gt + initial velocity Vi
height h = -gt^2/2 + Vi t + Hi the initial height

Answer 2

Ohh; I see. Thanks!