Question

Differentiate \[\cos^{-1}(e^x)\] with respect to \[x\].

Options

• \[\frac{e^x}{\sqrt{1-e^{2x}}}\]

• \[\frac{-e^x}{\sqrt{1-e^{2x}}}\]

• \[\frac{-1}{\sqrt{1-e^{2x}}}\]

• \[\frac{-e^{2x}}{\sqrt{1-e^x}}\]

Answer 1

Use \[\frac{d}{dx}(\cos^{-1}u)=\frac{-1}{\sqrt{1-u^2}}\cdot\frac{du}{dx}\] with \[u=e^x\]. Because \[\frac{d}{dx}(e^x)=e^x\], the result is \[\frac{-e^x}{\sqrt{1-e^{2x}}}\].