Over complicated compared to classical numerical methods.Originally Posted by mubashirmansoor
Hello, The latest research of mine concerning the derivative of a function without using diffrentiation and even without having the knowledge of what the equation of the funtion is, is attached to the post, please send me your comments on this latest document of mine.
Well Captain, It is complicated compared with diffrentiation But the good thing about it is that we dont need to know the function with which the terms are related, The main Uses of this formula is to find the accleration of the heavenly bodies etc...
Now what did you really mean by numerical methods? Do we needto know the polynomial for finding the gradient????
Thanks for Your Attention.
Well yes but this formula is applicable to all curves while there are diffrent types of the diffrentiated formula you are talking about for diffrent graphs...
For clarification and more accuracy I've done some changes written in the attached file.
The ultimate formula is;
dy/dx =10^10*tan (½(tan-1((f(x+1)–f(x))/10^10)+tan-1((f(x)–f(x-1))/10^10)))
No it is exact for 2-nd degree curves only. Also I beleive it is identical inOriginally Posted by mubashirmansoor
practice with what I gave above.
Try it for at .
Asking for what numerical value your method gives for your example is
to indicate to you that the calculation is not simple. For instance how
many transendental function evaluations are involved? (transendental
functions include the trig functions and their inverses, as well as logs
and exponential functions).
Sorry for the delay in my response,
You have written that the answers are not accurate in the formula provided, There is something happening which I don’t understand, (I’m a student of O’levels so I’d be glad if you help me figure out the problem) if we use the formula with small gaps between the terms, it works very accurately & there is just .00000001 difference in the answers compared to differentiation. I mean the formula given below;
dy/dx = tan ( 0.5 ( tan-1 (( f(x+0.1) – f(x) ) / 0.01) + tan-1 (( f(x) – f(x-0.01) / 0.01)))
This equation is even applicable for x^20 or higher degree equations… To get rid of inserting these extremely nearby points, I thought to decrease the size of the curve (not the dimensions) by 10^10, in this way 1 unit would turn to 10 billionth of a unit and even the points with huge gaps between each other would become extremely near. So when the calculation was taking place the curve was small enough to get good approximations, and after the evaluation the curve was brought to its original size hence multiplied by 10^10.
I don’t feel that the technique has a problem but its answers are absolutely the same as differentiation for degree 2 equations but as the degrees increase the answers become larger and larger by about 1 to 3 units for the degrees 3 and 13/15 respectively.
We know that differentiation is itself not accurate and that we say “delta x” approaches zero which is impossible, and the real answers should be a little bit more than that of differentiation. What’s the problem with the last part of my technique???
I’ll be really thankful if you help me figure out what’s going on…
Differentiation is exact, it gives the exact value of the slope at a point on a curve.Originally Posted by mubashirmansoor
You will have to learn some calculus before you will understand what
is going on in detail.
A good (cheap-ish) book to learn calculus from is "Calculus: An Intuitive and
Physical Approach" by Morris Kline (see here )
Then a good book to learn some numerical analysis from is "Numerical
Methods that Work" by Forman Acton (see here )
is used as a stage in the development of the derivative.Originally Posted by mubashirmansoor
The derivative is the slope of a curve at a point, we approximate this by
calculating the slope of a secant of the curve near the point of interest.
Then by letting the points at which the secant meets the curve near the
point both approach the point of interest in the limit obtain the exact slope
of the curve at the point if this should happen to exist. The slope of the
secant is .
If you consult a reference on calculus as I have suggested earlier you will
learn what is going on better than you will form a discussion here.
The link provided is to be used for having a look at the detailed discusion done on this topic;