Find the derivative of $f(x) = \arcsin(e^x)$.

Determine the domains of the function and its derivative.

$ \begin{equation} \begin{aligned} f'(x) &= \frac{1}{\sqrt{1 - (e^x)^2}} - \frac{d}{dx} (e^x)\\ \\ f'(x) &= \frac{e^x}{\sqrt{1-e^{2x}}} \end{aligned} \end{equation} $

$ \begin{equation} \begin{aligned} \text{Since the domain of the inverse sine function is } [-1,1]\text{ , the domain of } f \text{ is } - \leq e^x \leq 1 &= \ln (-1) \leq x \ln e \leq \ln 1\\ \\ &= \infty \leq x \leq 0\\ \\ &= (\infty, 0, ] \end{aligned} \end{equation} $

The whole domain of $f'(x)$ is...

$ \begin{equation} \begin{aligned} &= 1 - e^{2x} > 0\\ \\ &= 1 > e^{2x}\\ \\ &= \ln (1) > (\ln e) (2x)\\ \\ &= 0 > x \end{aligned} \end{equation} $

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