Very basic derivative question Why is the derivative of 
$e^x + e^{-x}$
equal to
$e^x - e^{-x}$
?

By the chain rule, the derivative of $e^{f(x)}$ is $f'(x)e^{f(x)}$. Hence, $$\frac{d}{dx}[e^{x} + e^{-x}] = \left[\frac{d}{dx}(x)\right]\cdot e^{x} + \left[\frac{d}{dx}(-x)\right]\cdot e^{-x} = e^{x} - e^{-x}$$
