Question

If $y=(1+x)(1+x^2)(1+x^4).........(1+x^{2n})$ then the value of $\left(\frac{dy}{dx}\right)$ at $x=0$ is

• A. $0$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

The correct option is C $1$

Take log on both the sides
$\log y=\log (1+x)+\log (1+x^2)+\log (1+x^4)+.........+\log (1+x^{2n})\frac{1}{y}\frac{dy}{dx}=\frac{1}{1+x}+\frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+........+\frac{2n{x}^{2n-1}}{1+x^{2n}}\frac{dy}{dx}=y(\frac{1}{1+x}+\frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+........+\frac{2n{x}^{2n-1}}{1+x^{2n}})$
${\left(\frac{dy}{dx}\right)}_{x=0}=1$