What is the derivative of $\displaystyle e^{-u}$ ???
It seems so simple yet I always struggle with it.
You've neglected an extremely important piece of information .... derivative of $\displaystyle e^{-u}$ with respect to what variable? u? x?
The derivative of $\displaystyle e^{-u}$ with respect to u is $\displaystyle -e^{-u}$.
The derivative of $\displaystyle e^{-u}$ with respect to x is $\displaystyle -e^{-u} \, \frac{du}{dx}$.