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

Answer 1

y=(1+x)(1+x²)(1+x⁴)......(1+x²ⁿ)

⇒log y =Iog (1+x)+ Iog (1+x²) ......+ Iog (1+x²ⁿ)
⇒1/y. dy/dx= 1/(1+x )+ 2x/(1+x² )......+ 2nx²ⁿ⁻¹/(1+x²ⁿ)

⇒dy/dx= y.(1/1+x + 2x/1+x².......+ 2nx²ⁿ⁻¹/1+x²ⁿ)
Substitute the value x=0,
⇒∣dy/dx∣_x=0 =1.1 = 1