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