Taylor expansion of $e^{\cos x}$ I have to find the 5th order Taylor expansion of $e ^{\cos x}$. I know how to do it by computing the derivatives of the function, but the 5th derivative is about a mile long, so I was wondering if there is an easier way to do it.
I'd appreciate any help.

You can start with the Taylor expansion of $\cos x$ and when you expand 
$\exp(\cos x)$, you just throw away terms you know won't affect the final result.
For lower order Taylor expansion, the derivation is actually pretty short and straight forward.
$$\begin{align}
\exp(\cos x)
&= \exp\left(1 - \frac{x^2}{2} + \frac{x^4}{24} + O(x^6)\right)\\
&= e\left[1 + \left(-\frac{x^2}{2} + \frac{x^4}{24} + O(x^6)\right) + \frac{1}{2}\left( -\frac{x^2}{2} + O(x^4)\right)^2 + O(x^6) \right]\\
&= e\left[ 1 - \frac{x^2}{2} + \left(\frac{1}{24} + \frac{1}{8}\right) x^4 \right] + O(x^6)\\
&= e\left[ 1 - \frac{x^2}{2} + \frac{x^4}{6} \right] + O(x^6)
\end{align}
$$
