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.

Use 
$$e^{\cos x}=e\cdot e^{\cos x-1}.$$
Then substitute the power series expansion of $\cos x-1$ for $t$ in the power series expansion of $e^t$. What makes this work is that the series for $\cos x-1$ has $0$ constant term. For terms in powers of $x$ up to $x^5$, all we need is the part $1+t+\frac{t^2}{2!}$ of the power series expansion of $e^t$, and only the part $-\frac{x^2}{2!}+\frac{x^4}{4!}$ of the series expansion of $\cos x -1$. 
