Question

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

Answer 1

log y = log(1 + x) (1 + x²) (1 + x⁴).....(1 + x²ⁿ)

log y = log(1 + x) + log(1 + x²) + log(1 + x⁴) + ....+log(1 + x²ⁿ)

\(\frac1y\frac{dy}{dx} = \frac1{1+x}+\frac{2x}{1+x^2} + \frac{4x^3}{1+x^4}+....+\frac{2n x^{2n-1}}{1+x^{2n}}\) 

y at x = 0 is y(0) = 1

\(\therefore\) \(\frac{dy}{dx}\) at x = 0

\(\therefore\) (\(\frac{dy}{dx}\))_{(x = 0)} = y(0)(\(\frac1{1+0}+\frac{2\times0}{1+0}+\frac{4\times0}{1+0}\).....+\(\frac{2n-0}{1+0}\))

 = 1( 1 + 0 + 0 + .... + 0) = 1