Single Variable Calculus, Chapter 7, 7.6, Section 7.6, Problem 36

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}
$