Single Variable Calculus, Chapter 7, 7.3-2, Section 7.3-2, Problem 42

Differentiate $\displaystyle y = e^{k \tan \sqrt{x}}$


$
\begin{equation}
\begin{aligned}

y' =& \frac{d}{dx} (e^{k \tan \sqrt{x}})
\\
\\
y' =& e^{k \tan \sqrt{x}} \frac{d}{dx} (k \tan \sqrt{x})
\\
\\
y' =& e^{k \tan \sqrt{x}} \left[ k \frac{d}{dx} (\tan \sqrt{x}) + (\tan \sqrt{x}) \frac{d}{dx} (k) \right]
\\
\\
y' =& e^{k \tan \sqrt{x}} \left[ k \sec^2 \sqrt{x} \frac{d}{dx} (x)^{\frac{1}{2}} + 0 \right]
\\
\\
y' =& e^{k \tan \sqrt{x}} k \sec^2 \sqrt{x} \cdot \frac{1}{2^{\frac{1}{2}}}
\\
\\
y' =& \frac{e^{k \tan \sqrt{x}} k \sec^2 \sqrt{x} }{2 \sqrt{x}}



\end{aligned}
\end{equation}
$