$\Phi(\cdot,0,1)$ 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 $w$?