Suppose I have $q(x)$ which represents the quantile function of a distribution. I can query the function for its value and derivative at any point (e.g. I can ask for $q(0.6)$ and $q'(0.221)$).
Is this enough to recover (approximate) $p(x)$, the pdf of the distribution associated with $q$? How might one go about doing this?
The quantile function is exactly the inverse function to the cumulative distribution function (assuming that this CDF is strictly increasing, at least), and the density function is the derivative of the CDF (assuming the CDF is differentiable). So $p(x) = \frac d{dx}(q^{-1}(x)) = 1/q'(q^{-1}(x))$.