It is easier to take the derivative, and consider the volume of the $(n-1)$-sphere (i.e., the "surface area" of the boundary of the ball).
Start with the integral $\int_{\mathbb{R}^n} e^{-x_1^2 - \cdots - x_n^2} dx_1 \cdots dx_n$. Fubini's theorem lets you decompose this into a product of $1$-dimensional integrals, and you get $\pi^{n/2}$. Since the integrand is spherically symmetric, you can change to the integral $\int_0^\infty \mathrm{vol}(S^{n-1}(r)) \, e^{-r^2} dr$, where $S^{n-1}(r)$ is the unit $(n-1)$-sphere of radius $r$. The volume of this sphere is $r^{n-1}$ times the volume of the unit sphere, so solving for that, you get $\frac{\pi^{n/2}}{\int_0^\infty r^{n-1} e^{-r^2} dr}$. A change of coordinates ($u = r^2$) in the denominator yields the integral defining $\Gamma(n/2)$.
