How to find dy/dx by logarithmic differentiation

Question

The question says find dy/dx by logarithmic differentiation 2 Definite integral. E^-1/x divided by x^2 dx 1

Answer choices are

A 1-sqrt(e)/e

B 1-e

C sqrt(e)-1/e

D sqrt(e)-e/e

E sqrt(e)

If you could provide an explanation and/or show steps it would be greatly appreciated.

Answer 1

Hint: Let me outline the general procedure of logarithmic differentiation. $$y=f(x)\\ \implies \dfrac{y^\prime}{y}=\dfrac{d}{dx}\ln(f(x))\\ \implies y^\prime=f(x)\times \dfrac{d}{dx}\ln(f(x))$$ You need to also use the fundamental theorem of calculus in this problem.