How can I calculate the derivative of the standard normal inverse. I think the derivative of $\Phi^{-1}(x)$ is $$\frac{1}{\phi(\Phi^{-1}(x))}.$$ I would like to know how to find the derivative of $$\Phi^{-1} \left(\frac{x}{c}\right),$$ where $c$ is a known constant.
Any help would be much appreciated.
Best Answer
Apply the chain rule:
$$\frac{d}{dx}\left(\Phi^{-1}\left(\frac{x}{c}\right)\right)=\frac{d}{dx}\Phi^{-1}{\bigg|}_{\frac{x}{c}}\cdot\frac{d}{dx}\left(\frac{x}{c}\right)=\frac{1}{\Phi'(\Phi^{-1}(\frac{x}{c}))}\cdot\frac{1}{c}$$