  For a twice continuously differentiable function $S$, we define the density
function of its gradient (derivative in one dimension) $s = S^{\prime}$ as a
random variable transformation of a uniformly distributed random variable using
$s$ as the transformation function. Given $N$ values of $S$ sampled at equally
spaced locations, we demonstrate using the method of stationary phase that the
approximation error between the integral of the scaled, discrete power spectrum
of the wave function
$\phi^{D}_{\tau}=\frac{1}{\sqrt{L}}\exp\left(\frac{iS}{\tau}\right)$ and the
integral of the true density function of $s$ over an arbitrarily small interval
is bounded above by $O(1/N)$ as $N \rightarrow \infty$ ($\tau \rightarrow 0$).
In addition to its easy implementation and fast computability in $O(N \log N)$
that only requires computing the discrete Fourier transform, our framework for
obtaining the derivative density does not involve any parameter selection like
the number of histogram bins, width of the histogram bins, width of the kernel
parameter, number of mixture components etc. as required by other widely
applied methods like histograms and Parzen windows.
