Suppose that $X$ is a random variable for which the p.d.f. or the p.f. is $f(x|\theta)$, where the value of the parameter $\theta$ is unknown but must lie in an open interval $\Omega$. Let $I_0(\theta)$ denote the Fisher information in $X.$ Suppose now that the parameter $\theta$ is replaced by a new parameter $\mu$, where $\theta = \psi(\mu)$ and $\psi$ is a differentiable function. Let $I_1(\mu)$ denote the Fisher information in $X$ when the parameter is regarded as $\mu.$ Show that $$I_1(\mu) = [\psi'(\mu)]^2 I_0[\psi(\mu)].$$
How would I do this? Do I need to use a Taylor expansion? Regardless, I would appreciate a written proof. This isn't for class but the above statement has been mentioned in texts without any detail whatsoever.