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$\Phi(\cdot,0,1)$ and $\phi(\cdot,0,1)$ are cdf and pdf of standard normal distribution. $$y=F_\text{mix}(x,\mu,\sigma)=\sum\limits_{i=1}^{K}\lambda_i\Phi\left(\frac{x-\mu_i}{\sigma_i},0,1\right).$$

$x=Q(y)$ is the inverse function of $F_\text{mix}$.

$$\mu_i=\bar{\mu_i}'w,\qquad \sigma_i^2 =w'\Sigma_i w.$$

What is the derivative of $Q(y)$ with respect to $w$?

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