Answer
$$
f(x)=\int_{1}^{\sqrt{x}} \frac{e^{s}}{s} d s
$$
$\Rightarrow $
$$
f^{\prime}(x)=\frac{e^{\sqrt{x}}}{2 x}
$$
Work Step by Step
$$
f(x)=\int_{1}^{\sqrt{x}} \frac{e^{s}}{s} d s
$$
$\Rightarrow $
$$
\begin{aligned}
f^{\prime}(x)&=\frac{d}{d x} (f(x))\\
&=\frac{d}{d x} \int_{1}^{\sqrt{x}} \frac{e^{s}}{s} d s \\
& \quad\quad\text { Let } u=\sqrt{x} . \text { Then } \frac{du}{dx} =\frac{1}{2 \sqrt{x}}\\
&=\frac{d}{d u} \left[ \int_{1}^{u} \frac{e^{s}}{s} d s \right] \frac{du}{dx} \quad\quad\text { (by the Chain Rule) }\\
&=\frac{e^{u}}{u}\frac{du}{dx} \quad\quad\quad\quad \quad\quad \text { (by FTC1)}
\\
&=\frac{e^{\sqrt{x}}}{\sqrt{x}} \frac{1}{2 \sqrt{x}} \\
&=\frac{e^{\sqrt{x}}}{2 x}
\end{aligned}
$$
