Calculus

Well its been a long time since i did some real mahs.. Besides i promised Teck Siang. So here it is.

y=f(x)
dy/dx=f '(x) (1)

On the other hand,
y=f(x)
F(y)=x , where F is the inverse function of f.
F '(y)=dx/dy
dy/dx=1/F '(y)
dy/dx=1/F '[f(x)] (2)

Combining equations (1) and (2), we have

f '(x)=1/F '[f(x)]
F '[f(x)]=1/f '(x)

Well most of you (including myself when i came up with this) must be thinking, wtf is that, and how can something like that actually be useful?

Well my original intention was to find an equation to relate the inverse function with its derivative. Which would have made me famous overnight..XD Sadly, this cannot be manipulated to give me such an equation, but it does have interesting uses.

for eg to find the integral of 1/x, we let f '(x)=1/x
then, f(x) must be the integral.

F '[f(x)]=1/f '(x)
F '[f(x)]=1/1/x=x
since the equation is an identity in x, we can replace every x in the equation with F(x)

F '[f(F(x))]=F(x)
since F is the inverse of f, f[F(x)] must be equal to x
F '(x)=F(x)
Now think about what this equation is trying to say. Its simply telling you that the derivative of F(x) is equal to itself. In fact we only know one such function , which is e^x. Haha..

So since F(x)=e^x
f(x) must be equal to the inverse of e^x, which is lnx.
f(x)=lnx

Thus we have shown, without doing ANY differentiation or inteegration of both 1/x and lnx, that the integral of 1/x is lnx. Quite cool. Don't you think? In fact the only two important properties we have used are,

1/dy/dx=dx/dy
and that
e^x is its own derivative.

Sadly, this "trick" that i've used doesn't work for other functions.. Maybe i can develop it further.. hmm...

In any case, I think either i'm going bonkers, otherwise I'm getting smarter.

A little sidenote: A lot of people coming to my site tell me they don't understand the mathematics on my site. In fact, that is perfectly normal. Even for professional mathematicians, they do not understand new mathematics the first time they read it. Most of the time, hey have to analyze and go through the text many times before they can fully grasp it. This is where the problem with many people lie. When they do not understand the text, they do not take the effort to properly anaylyze or think about the text. Just like music and art and literature, effort must be made to understand the subject before one can fully appreciate its beauty. And i promise you, the mathematical beauty that we speak of is certainly real. (with the exception of complex mathematics, where things are imaginary XD)

In fact i find it a waste that people get put off by mathematics because it seems so hard to grasp. Mathematics is a language more useful than any other.