Differentiating e to the cos x from the limit definition

Using only the definition of the derivative as a limit, find the derivative of $g(x)=e^{\cos x}$.

$g'(x)=-\sin x\,e^{\cos x}$. Write $\frac{g(x+h)-g(x)}{h}=e^{\cos x}\cdot\frac{e^{\Delta}-1}{\Delta}\cdot\frac{\Delta}{h}$ with $\Delta=\cos(x+h)-\cos x$. As $h\to0$, $\Delta\to0$ so the middle factor tends to $1$, and $\Delta/h=-\sin x\frac{\sin h}{h}-\cos x\frac{1-\cos h}{h}\to-\sin x$.