Why are derivatives important in math?

Answer 1

It is a measure of the rate of change of one variable - relative to another. The measure is an instantaneous measure rather than one averaged over a longer domain. Such changes are fundamental to many real-life events.

Answer 2

There are lots of situations in physics where one quantity is the derivative of another one - for example, velocity is the derivative of position, acceleration is the derivative of velocity, power is the derivative of energy, etc.One extremely useful application of derivatives is to find a maximum or a minimum of a function. Such a maximum or minimum may happen at one of the following points:

* The endpoints of the interval you are considering,

* A point where the derivative is zero, or

* A point where the derivative is undefined.

So, to calculate a maximum or minimum, you will usually need to calculate the derivative.