Question

What is the derivative of `ln x / e^x`?

Answer 1

First, we can rewrite this expression as a product instead of a quotient:

`ln(x)e^-x`

And, now, remembering the product rule:

Be `y=f(x)g(x)`, then `(dy)/(dx)=f'(x)g(x)+f(x)g'(x)`, we'll get

`(dy)/(dx)=1/x*e^-x+ln(x)(-e^-x)`

`(dy)/(dx)=color(green)(e^-x(x^-1+ln(x)))`