<s>[INST] Differentiate the Function: $y=e^{k\ tan\sqrt{x}}$ $y=e^{k\tan\sqrt{x}}$
$=e^{k\tan\sqrt{x}}\cdot [{k\tan\sqrt{x}}]'$
$=e^{k\tan\sqrt{x}}\cdot\ (k)\cdot[\tan x^{\frac{1}{2}}]'+(\tan x^{\frac{1}{2}})\cdot[k]'$
$=e^{k\tan\sqrt{x}}\cdot\ (k)\cdot (\frac{1}{2}tanx)\cdot \sec^2x + (\tan x^{\frac{1}{2}})\cdot (1)$
Is my process correct? 
 [/INST] Your work is not quite correct.
$$\begin{align}
(e^{k\tan\sqrt x})' &= e^{k\tan\sqrt x}\cdot(k\tan\sqrt x)' \\
 &= e^{k\tan\sqrt x}\cdot k\cdot (\tan\sqrt x)' \\
 &= e^{k\tan\sqrt x}\cdot k\cdot\sec^2\sqrt x\cdot (\sqrt x)' \\
 &= e^{k\tan\sqrt x}\cdot k\cdot\sec^2\sqrt x\cdot \frac 1{2\sqrt x} \\
 &= \frac{k\sec^2\sqrt x\cdot e^{k\tan\sqrt x}}{2\sqrt x}
\end{align}$$
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