Question

How do you differentiate `f(x)= e^x/(e^(-x) -2)` using the quotient rule?

Answer 1

`f'(x) = 2frac{1-e^x}{(e^{-x}-2)^2}`

Explanation:

`f'(x) = frac{d}{dx}(frac{e^x}{e^{-x}-2})`

`= frac{(e^{-x}-2)frac{d}{dx}(e^x)-e^xfrac{d}{dx}(e^{-x}-2)}{(e^{-x}-2)^2}`

`= frac{(e^{-x}-2)e^x-e^x(-e^{-x})}{(e^{-x}-2)^2}`

`= frac{1-2e^x+1}{(e^{-x}-2)^2}`

`= 2frac{1-e^x}{(e^{-x}-2)^2}`