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 $\int\_{\mathbb{R}^n} e^{-x\_1^2 - ... - x\_n^2} dx\_1 \dots dx\_n$. 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 $\int\_0^\infty vol(S^{n-1}(r)) e^{-r^2} dr$, 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 = r2) in the denominator yields the integral defining $\Gamma(n/2)$.