Proof of derivative of inverse
- From: ikita <nevillelast@xxxxx>
- Date: Sun, 07 May 2006 18:51:11 EDT
Hi!
I'm new here and I apologize if my posting is in the wrong place. :)
I did a calculus exam a while ago and one question was "Define and prove the derivative of the inverse of a function". I thought I could prove it by using Liebniz (?) notation dy/dx => inverse: dx/dy but then I used a long proof involving the definition of derivatives (lim (h->0) (f(x+h)-f(x))/h) that follows below.
I now know there is a proof: y=f(x) and y+k = f(x+h) => h=(f^-1)(y+k)-(f^-1)(y) => h/k=((f(x+h)-f(x))/h)^-1 since h->0 when k->0.
However my proof was way longer and much dumber (if even correct). What I'm asking is whether my proof is correct or not - (and if so, why)? Here it is:
x(f) is the invese to f(x) and h->0:
x(f)*f(x)+x(f+h)*f(x+h) = f(x)*x(f+h)+x(f)*f(x+h) =>
Left-Right = x(f)*f(x)+x(f+h)*f(x+h)-f(x)*x(f+h)-x(f)*f(x+h) = (x(f+h)-x(f))(f(x+h)-f(x)) = 0 = h^2 =>
h/(f(x+h)-f(x)) = x(f+h)-x(f))/h = 1/f'(x) = x'(f) Q.E.D.
My teacher gave my zero points and wrote I had used two different h - I don't understand, please help. I have a feeling something major is wrong with my proof but I'll leave it to the experts (you). :)
Thanks for all answers!
.
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