Differentiate the function cos^-1(e^x) with respect to x.

The correct answer is Option (3) → $\frac{-e^x}{\sqrt{1-e^{2x}}}$ ##

Let $y = \cos^{-1}(e^x)$.

Using chain rule, we have

$\frac{dy}{dx} = \frac{-1}{\sqrt{1-(e^x)^2}} \cdot \frac{d}{dx}(e^x) = \frac{-e^x}{\sqrt{1-e^{2x}}}$