what is differenciation ?Please describe it briefly.
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In mathematics, process of finding the derivative, or rate of change, of a function. In contrast to the abstract nature of the theory behind it, the practical technique of differentiation can be carried out by purely algebraic manipulations, using three basic derivatives, four rules of operation, and a knowledge of how to manipulate functions.
I'll give you a brief idea of what differentiation means. Maybe someone else can go more in depth..
Let's look at a physical example. Suppose you're driving around at 50 km/h (if you're more comfortable with the imperial system, let's say 50 mph then, but whatever, moot point). If your velocity doesn't change, then after an hour you would have traveled 50 km. Conversely, if you traveled 50 km in one hour, you can say that you traveled 50 km/h provided that you moved at a constant velocity.
But what if you didn't travel at a constant velocity? Seems much more likely.. But you want to find out how fast you traveled at a particular time, say at the 30 min mark, and not one billionth of a second earlier or later. How do we do that?
Well, to solve that problem is the idea behind what differentiation is and the velocity at a time is the derivative of the position at the same. To find the derivative of position or to find the velocity, what you would do is pick a time before the 30 min mark and get the position there. Then pick a time after the 30 min mark and get the position there. Then take the average of the two. That gives you the average velocity. But we want the instantaneous velocity. Well, then let's pick a time closer to the 30 min point on both sides. That will give a better result. And we continue and we continue forever. The limiting process of this is then the derivative.
We can also get the acceleration which is the derivative of the velocity. So assuming we knew the velocity, we do the same thing. We need the velocity at a particular time before the 30 min mark and a particular time after and we let them get closer and closer to the 30 min mark.
There are many more interesting examples in nature. For example, Faraday's law says that the induced voltage on a coil is related to the derivative of the magnetic flux through the coil.
I'll give you a brief idea of what differentiation means. Maybe someone else can go more in depth..
Let's look at a physical example. Suppose you're driving around at 50 km/h (if you're more comfortable with the imperial system, let's say 50 mph then, but whatever, moot point). If your velocity doesn't change, then after an hour you would have traveled 50 km. Conversely, if you traveled 50 km in one hour, you can say that you traveled 50 km/h provided that you moved at a constant velocity.
But what if you didn't travel at a constant velocity? Seems much more likely.. But you want to find out how fast you traveled at a particular time, say at the 30 min mark, and not one billionth of a second earlier or later. How do we do that?
Well, to solve that problem is the idea behind what differentiation is and the velocity at a time is the derivative of the position at the same. To find the derivative of position or to find the velocity, what you would do is pick a time before the 30 min mark and get the position there. Then pick a time after the 30 min mark and get the position there. Then take the average of the two. That gives you the average velocity. But we want the instantaneous velocity. Well, then let's pick a time closer to the 30 min point on both sides. That will give a better result. And we continue and we continue forever. The limiting process of this is then the derivative.
We can also get the acceleration which is the derivative of the velocity. So assuming we knew the velocity, we do the same thing. We need the velocity at a particular time before the 30 min mark and a particular time after and we let them get closer and closer to the 30 min mark.
There are many more interesting examples in nature. For example, Faraday's law says that the induced voltage on a coil is related to the derivative of the magnetic flux through the coil.
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