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The derivative of a function is a concept of differential calculus that characterizes the rate of change of a function at a given point. It is defined as the limit of the ratio of the function's increment to the increment of its argument when the argument's increment tends to zero, if such a limit exists. A function that has a finite derivative (at some point) is called differentiable (at this point).
The process of calculating the derivative is called differentiation. The reverse process - finding the original - integration.
At first glance, derivatives are needed to fill the heads of already overloaded schoolchildren, but this is not the case. Consider a car that drives around the city. Sometimes it stands, sometimes it drives, sometimes it brakes, sometimes it accelerates.
Let's say it drove 3 hours and drove 60 kilometers. Then, using the formula from elementary school, we divide 60 by 3 and say that she was driving at 20 km / h. Are we right? Well, partly right. We got the "average speed". But what's the use of it? The car can go at this speed for 5 minutes, and the rest of the time it either went slower or faster. What should I do?
And why do we need to know the speed for all 3 hours of the route? Let's divide the route into 3 parts for one hour and calculate the speed on each section. Let's. Let's say you get 10, 20 and 30 km/h. Here. The situation is already more clear - the car was driving faster in the last hour than in the previous ones.
But this is again on average. What if it just drove slowly for half an hour in the last hour, and then suddenly accelerated and started driving fast? Yes, it may be so.
As we can see, the more we break down our 3-hour interval , the more accurate we will get the result. But we don't need a "more accurate" result - we need a completely accurate result. This means that time must be divided into an infinite number of parts. And the part itself - therefore will be infinitely small.
If we divide the distance that the car has traveled in our infinitesimal period of time by this time, we also get the speed. But no longer average, but"instant". And there will also be infinitely many such instantaneous speeds.
If you understand all of the above , then you understand the meaning of the derivative. A derivative is the speed at which something changes. For example, in our case, speed is the speed at which the "distance traveled" changes over time. Or maybe "the speed of temperature change with a change in longitude to the North". Or " the speed of disappearing sweets from a vase in the kitchen." In General, if there is something, a certain value "Y", which depends on some value "X", then most likely, there is a derivative that is written dy/dx. And it just shows how the value of y changes with an infinitesimal change in the value of x - how our distance changed with an infinitesimal change in time.