Derivative of a finite continued fraction

AvgStudent

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I got this as a challenge question from my professor, but not sure how to start. Let the fraction be be iterated 2018 times, and
f(x)=x1x1...1x1f(x)=x-\frac{1}{x-\frac{1}{...-\frac{1}{x-1}}}Find f(0)f'(0)
 
I got this as a challenge question from my professor, but not sure how to start. Let the fraction be be iterated 2018 times, and
f(x)=x1x1...1x1f(x)=x-\frac{1}{x-\frac{1}{...-\frac{1}{x-1}}}Find f(0)f'(0)
Since this is a challenge question - you need to show more effort !!

I see a recursive relationship here.
 
Since this is a challenge question - you need to show more effort !!

I see a recursive relationship here.
With your clue:
fn+1(x)=x1fn(x)f_{n+1}(x)=x-\frac{1}{f_{n}(x)}However, I'm unsure about the first iteration. Is it f1(x)=xf_1(x)=x or f1(x)=x1f_1(x)=x-1?
 
Next, I tried to compute the derivative.
fn+1(x)=ddx(x1fn(x))=1+fn(x)fn(x)2f'_{n+1}(x)=\frac{d}{dx}\Bigg(x-\frac{1}{f_{n}(x)}\Bigg)=1+\frac{f'_n(x)}{f_n(x)^2}Is this correct? If so, how do I evaluate at 0?
 
Looks good to me.

But: don't be afraid to make mistakes. Move on, even if you are not 100% sure, and see where it gets you. Making mistakes is a very efficient way to learn.
 
So I tried writing out the first few terms for fn(0)f_n(0).
f1(0)=01=1f2(0)=0101=1f3(0)=1f4(0)=1f_1(0)=0-1=-1\\ f_2(0)=0-\frac{1}{0-1}=1\\ f_3(0)=-1\\ f_4(0)=1\\I see a pattern. For odd n, then fn(0)=1f_n(0)=-1. Whereas for even n, fn(0)=1f_n(0)=1.


As far as fn(0)f'_n(0), I also tried writing out the first few terms.
f1(0)=1f2(0)=1+1(1)2=2f'_1(0)=1\\ f'_2(0)=1+\frac{1}{(-1)^2}=2f3(0)=1+212=3f4(0)=1+3(1)2=4f'_3(0)=1+\frac{2}{1^2}=3\\ f'_4(0)=1+\frac{3}{(-1)^2}=4\\
 
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Looks to me that you are almost done.
 
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Time to change to "AboveAvgStudent" ? :)
Lol. I'm still on the fence about the answer because the question has 2018 iterations i.e n=2018. So are we looking for f2019(0)orf2018(0)?f'_{2019}(0) \quad or\quad f'_{2018}(0)?
 
It depends on whether x1x-1 is considered 0-th or 1st iteration :) Seriously, I don't know how to interpret the text of the problem, but I hope your teacher appreciate the solution without nitpicking on exact number of iterations.
 
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