Even analytic functions descend through squaring

Let \(f\) be even and analytic on a disk \(B(0,r)\). Define \[ g(z)=f(\sqrt z). \] Show that \(g\) is well-defined and analytic on \(B(0,r^2)\), and express \(g^{(n)}(0)\) in terms of derivatives of \(f\) at \(0\).