Question

What is the derivative of ln e^x / e^x -1

Answer 1

I assume you mean

$\dfrac{\mathrm d}{\mathrm dx}\ln\dfrac{e^x}{e^x-1}$

The chain rule says this is equivalent to

$\dfrac{\dfrac{\mathrm d}{\mathrm dx}\dfrac{e^x}{e^x-1}}{\dfrac{e^x}{e^x-1}}$

The numerator can be worked out via the quotient rule:

$\dfrac{\mathrm d}{\mathrm dx}\dfrac{e^x}{e^x-1}=\dfrac{e^x(e^x-1)-e^x\times e^x}{(e^x-1)^2}=-\dfrac{e^x}{(e^x-1)^2}$

We're left with

$\dfrac{-\dfrac{e^x}{(e^x-1)^2}}{\dfrac{e^x}{e^x-1}}=-\dfrac1{e^x-1}=\dfrac1{1-e^x}$