derivative of x/(x + x/(x + ...

[Stochastic_Jimmy]

I'm trying to figure out how to find the find the derivative of \(\displaystyle f(x) = \frac{x}{x + \frac{x}{x + \frac{x}{x + \dots}}} \).

My thought is that I need to find some clever way of rewriting \(\displaystyle f(x) \) so that it becomes relatively straightforward to find \(\displaystyle f'(x) \), but I can't think of another way of expressing it. One thought I had was to try to write it as some sort of limit, but I can't see how to make that work.

Any hints?

Thanks in advance!

 

[Deleted member 4993]

I'm trying to figure out how to find the find the derivative of \(\displaystyle f(x) = \frac{x}{x + \frac{x}{x + \frac{x}{x + \dots}}} \).

My thought is that I need to find some clever way of rewriting \(\displaystyle f(x) \) so that it becomes relatively straightforward to find \(\displaystyle f'(x) \), but I can't think of another way of expressing it. One thought I had was to try to write it as some sort of limit, but I can't see how to make that work.

Any hints?

Thanks in advance!

\(\displaystyle y \ = \ \dfrac{x}{x \ + \ y}\)

 

[Stochastic_Jimmy]

\(\displaystyle y \ = \ \dfrac{x}{x \ + \ y}\)

Ah, yes. That's clever! So then


\(\displaystyle yx + y^2 \ = \ x\)

and then I can just use implicit differentiation.

Thanks a lot!

 

[stapel]

By way of explanation....

...find the derivative of \(\displaystyle f(x) = \frac{x}{x + \frac{x}{x + \frac{x}{x + \dots}}} \).

Assume that the continued fraction has some "nice" form; call it "y":

. . . . .\(\displaystyle y\, = \,\frac{x}{x\, +\, \frac{x}{x\, +\, \frac{x}{x\, +\, \dots}}} \)

Note that, by nature of "infinity", the continued fraction can be "broken" at any point, and still be the same "size", so:

. . . . .\(\displaystyle y\, = \,\frac{x}{x\, +\, \left(\frac{x}{x\, +\, \frac{x}{x\, +\, \dots}}\right)}\, =\, \frac{x}{x\, +\, y} \)

...and so forth.

 

[Deleted member 4993]

Ah, yes. That's clever! So then


\(\displaystyle yx + y^2 \ = \ x\)

and then I can just use implicit differentiation.

Thanks a lot!

Now you should be able to differentiate:

\(\displaystyle \displaystyle{y \ = \ x^{x^{x^x....Ad \ \ infinitum}}}\)