query: If $y= \frac1{1+x} + \frac2{1+x^2} + \frac4{1+x^4} + \dots + \frac{2^n}{1+x^{2^n}} $, then find $\frac{\mathrm dy}{\mathrm dx}$.
positive_passages.docid: 26446
positive_passages.text: Find the sum : $\displaystyle\sum_{i=0}^n \frac{2^i}{1+x^{2^{i}}}$
negative_passages.docid: 149355
negative_passages.text: find $\left( \frac{x}{x+y} \right)^{2007} + \left( \frac{y}{x+y} \right)^{2007}$
negative_passages.docid: 69172
negative_passages.text: $y'=$ ${-x+\sqrt{(x^2+4y)}}\over 2$ , $y(2)=-1$
negative_passages.docid: 2487
negative_passages.text: Integral $\int \frac{\mathrm{d}x}{\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}}$
negative_passages.docid: 617
negative_passages.text: Find θ: $β=\arctan\frac{(R+r)·\sin(θ)-r·\sin(θ(1+\frac{R}{r}))}{(R+r)·\cos(θ)-r·\cos(θ(1+\frac{R}{r}))}$
negative_passages.docid: 13085
negative_passages.text: Evaluate $ 1 + \frac{1}{3}\frac{1}{4}+\frac{1}{5}\frac{1}{4^2}+\frac{1}{7}\frac{1}{4^3}+\dots$
negative_passages.docid: 4816
negative_passages.text: How to solve $\frac 1{1.2.3}+\frac 1{2.3.4}+\frac 1{3.4.5}+...$
negative_passages.docid: 124248
negative_passages.text: $1+\frac {1}{4}(1+\frac {1}{4}) +\frac {1}{9}(1+\frac {1}{4} +\frac {1}{9})+....$
negative_passages.docid: 114861
negative_passages.text: equation $\frac{\frac{1}{x^2}-\frac{1}{y^2}}{\frac{1}{x^2}+\frac{1}{y^2}}$
negative_passages.docid: 72906
negative_passages.text: Is $\frac{\partial}{\partial x_1}=\frac{\partial}{\partial y_1}$ if $x_1=y_1$?
negative_passages.docid: 100995
negative_passages.text: For what $x$ : $2\sqrt{x}+\dfrac{1}{\sqrt{x+1}} \leq 2\sqrt{x+1}$
negative_passages.docid: 10753
negative_passages.text: Natural number solutions to $\frac{xy}{x+y}=n$ (equivalent to $\frac 1x+\frac 1y=\frac 1n$)
negative_passages.docid: 129394
negative_passages.text: understanding $\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$
negative_passages.docid: 113593
negative_passages.text: How can I prove that $\frac{n^2}{x_1+x_2+\dots+x_n} \le \frac{1}{x_1}+ \frac{1}{x_2} +\dots+ \frac{1}{x_n}$?
negative_passages.docid: 48154
negative_passages.text: Find the minimum value of $f(x,y,z)=\frac{x^2}{(x+y)(x+z)}+\frac{y^2}{(y+z)(y+x)}+\frac{z^2}{(z+x)(z+y)}$.
negative_passages.docid: 13072
negative_passages.text: If $x,y,z\gt 0$ and $xyz=1$ Then minimum value of $\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$
negative_passages.docid: 8711
negative_passages.text: Problems in the relation $a=v\frac{\mathrm{d}v}{\mathrm{d}x}$