Differentiation of Product Series

Question

How to differentiate?

$$\frac{d [ (1-x)\times(1-x^2)\times (1-x^3)....(1-x^n) ]}{dx}$$

I tried to construct an expanding Series and then differentiate but i failed.

Please help

Answer 1

Hint : See Logarithmic Differentiation

Answer 2

hint:

let $u=(1-x)(1-x^2)(1-x^3)....$ $$\ln u=\ln(1-x)+\ln (1-x^2)+....$$ $$\frac{1}{u}\times\frac{du}{dx}=-\frac{1}{1-x}+-2x\frac{1}{1-x^2}+.....+(-n\times x^{n-1})\frac{1}{1-x^n}$$

$$\frac{du}{dx}=u\times[-\frac{1}{1-x}+-2x\frac{1}{1-x^2}+.....+(-n\times x^{n-1})\frac{1}{1-x^n}]$$

Thus this is your anaswer