I'll present to you a basic explanation on why it is so important and then provide a simple example. 

Knowing derivatives of a function, or the integral, is equivalent to knowing the general structure of it. Moreover, sometimes one could glue together local assumptions to get a full depiction of what we are after. Generally, local structures are easy to compute and thus feasible to compute!

Consider you'd like to know whether a car will hit you. In a bunch of seconds your brain detects whether it is braking or accelerating i.e. decreasing or increasing the speed. Which are the second derivative of the car's position! If the second derivative is high, then the car is still increasing its speed hence trying to hit you, otherwise it'll stop before even touching you. Seems a good enough reason to study analysis, doesn't it?
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