If $y= \frac1{1+x} + \frac2{1+x^2} + \frac4{1+x^4} + \dots + \frac{2^n}{1+x^{2^n}} $, then find $\frac{\mathrm dy}{\mathrm dx}$.

Question

If $y= \dfrac1{1+x} + \dfrac2{1+x^2} + \dfrac4{1+x^4} + \dots + \dfrac{2^n}{1+x^{2^n}} $, then find $\dfrac{\mathrm dy}{\mathrm dx}$.

I am stuck up in this question. I tried taking log on both sides and generate some simplified expression but was unable to do so. Any help would be highly appreciated. thanks

First read how to ask a good question then learn about $\LaTeX$, if you don't want to read here's another tutorial, though it's introduction does not apply to this site. . or you'll get downvotes. — UmbQbify, Aug 05 '20 at 05:59
I'm confused. Did the question specifically require that $dy/dx$ be simplified? If not, then, since the derivative of the sum = the sum of the derivatives, you could simply differentiate each fraction separately, and let the unsimplified result be your final answer. — user2661923, Aug 05 '20 at 06:12

score 1 · Answer 1 · answered Aug 05 '20 at 05:59

1

Hint:

$$\dfrac1{1+y}+\dfrac1{1-y}=\dfrac2{1-y^2}$$

Set $y=x,x^2,x^4,\cdots,x^{2^n}$ to find

$$y+\dfrac1{1+x}=\dfrac{2^n}{1-x^{2^{n+1}}}$$

answered Aug 05 '20 at 05:59

lab bhattacharjee

274,582

It's derivative would simplify too but I'm unable to match up to the answer.. – user 34152 Aug 06 '20 at 04:16
@user55124, What's the answer – lab bhattacharjee Aug 06 '20 at 06:49

If $y= \frac1{1+x} + \frac2{1+x^2} + \frac4{1+x^4} + \dots + \frac{2^n}{1+x^{2^n}} $, then find $\frac{\mathrm dy}{\mathrm dx}$.

1 Answers1