Upper Bound of Difference of Exponentials

Question

I know that the upper bound of an exponential function is given by $e^{-x} \leq \frac{1}{1+x}$, but how do I find the upper bound of the difference of exponential functions?

For example, what is the upper bound of $e^{-a} - e^{-b}$ ?

Answer 1

You can use the Mean Value Theorem $$\frac{f(b)-f(a)}{b-a}=f'(\xi)$$ with $\xi \in [a,b]$