If `y = e^(x) * e^(x^(2)) *e^(x^(3) )*...*e^(x^(n))..., " for " - lt x lt 1 , " then" (dy)/(dx) ` at
` x = (1)/(2) ` is

Options

A. e
B. 4e
C. 2e
D. 3e

Answer

B

Explanation

Given , ` y = e^(x) *e^(x^(2)) * e^(x^(2)) * … e^(x^(n))*... `
`rArr y = e^(x)*e^(x^(2))*...e^(x^(2))*...`
` rArr (dy)/(dx) = e^((x)/(1-x)) [ ((1-x)1-x(-1))/((1-x)^(2))]= e^((x)/(1-x)).(1)/((1-x)^(2))`
At ` x = (1)/(2), ((dy)/(dx))_(= (1)/(2)). = e^((1//2)/(1-1//2))= 4e `