Let $V_n$ be the volume of the $n$-dimensional unit hypersphere. A random variable $x$ is chosen from the hypersphere, where $x$ belongs to one of the hypercylinders/hyperdisk inside the hypersphere. The volume of the hyperdisk where $x$ belongs is given as $$V_{n-1}(\sqrt{1-x^2})^{n-1}.$$
Now the probability of $x$ can be calculated as the ratio of volume of the hyperdisk to the volume of the n-dimensional unit hypersphere. I want to calculate the probability density function of the $x$. I know that the derivative of probability of $x$ gives PDf of it. However, in one article, the PDF $x$ ($f_n(x)$) in this situation is given as $$f_n(x) \,\mathrm{d}x = \frac{V_{n-1}(\sqrt{1-x^2})^{n-1} \,\mathrm{d}x}{V_n}.$$
Is this relation correct? How can I derive this relation? Thank you.