$()$ Show that the softplus activation function $\backslash$zeta $($a$)$ given by $(6\mathrm{.}16)$ satisfies the

properties:

$\backslash$zeta $($a$)\backslash$zeta $($a$)=$ a $(6\mathrm{.}62)$

ln $\backslash$sigma $($a$)=\backslash$zeta $($a$)(6\mathrm{.}63)$

d$\backslash$zeta $($a$)$

da

$=\backslash$sigma $($a$)(6\mathrm{.}64)$

$\backslash$zeta

$1($a$)=$ ln $($exp$($a$)1)(6\mathrm{.}65)$

where $\backslash$sigma $($a$)$ is the logistic$-$sigmoid activation function given by $(6\mathrm{.}13).$