# What is the second derivative of f(x)=(e^x + e^-x) / 2 ?

$f ' ' \left(x\right) = f \left(x\right)$
$f \left(x\right) = \frac{1}{2} \left({e}^{x} + {e}^{-} x\right)$
$f ' \left(x\right) = \frac{1}{2} \left({e}^{x} - {e}^{-} x\right)$ $\text{ }$ $\left(\text{use } \frac{d}{\mathrm{dx}} \left({e}^{u}\right) = {e}^{u} \frac{\mathrm{du}}{\mathrm{dx}}\right)$
$f ' ' \left(x\right) = \frac{1}{2} \left({e}^{x} + {e}^{-} x\right) = f \left(x\right)$