diffrentiation

Well this is my understandin of Diffrentiation.... its easy if we understand this.. but the graphical method is undoubtably the best if thought properly.

x^2 when diffrentiated gives 2x.
Ha ha, where the hell do you think this will be used, I can use it more better only if I understand it well na!!
ok the understanding can be got simply by replacing by numbers say I replace 1000 instead of x.
so x^2 should be 1000000
ie
x = 1000 => x^2 = 1000000

Now assume I increase x by 1
so we have x = 1001 and so x^2 = 1002001,
ie
x = 1001 => x^2 = 1002001

so we see that the change is just most significantly 2*x* the change + a miniscle number
The number being more smaller the change becomes very prominent only on the 2x part.
so that is why we have
derivative of x^2 is 2x
Similarly we can get the description for x^3
and so on!!!


You may ask...
x = 1000.5 => x^2 = 1001000.25
which is not 2x!!! well that is why in the first principles of diffrentiation we do have to devide by the inc. and then ignore the very small value when compared to the large value. So that we can approximate the change at any place to be 2x.
ie the change is around 2*x*the inc!!! inc being small, and then normalizing using the small inc (very small but not zero!!!) we realize that the change is just 2 times x .... why normalize?? coz that will make it more generic rt!!! [:)]...said in other words... we have now made the change in the whole values as independent as possible of the minute changes!!!.


Well all this is just gimic.. the actual meaning of a derivative is the rate at which somethign is changing...
Lets understand it better by taking a line say y = x
the derivative is 1....
it means that there is no change of the rate at which the slope is changing.. its constant.
Now if we have a parabola say y = x^2
Lets see this
x 1 2 3 4 5 6 7 8

y 1 4 9 16 25 36 49 64

dX 1 1 1 1 1 1 1

dY 3 5 7 9 11 13 15

dY/Dx 3 5 7 9 11 13 15 -- we see that the values are all odd integers and can be represented by a straight line so the derivative of a parablola is a line..
It is at each of these minute intervals if we calulate the slope.. the rate at which the slope of the prabola changes we get that to be a straight line.