Why does integration by parts work?

Let's consider the form of the formula we use for integration by parts: the integral of u * dv/dx = uv - the integral of v * du/dx. We know that integration is the inverse of differentiation, so we should be able to differentiate both sides to get back to u * dv/dx. The left hand side of this equation obviously satisfies this. d/dx(uv - integral of v * du/dx) = d/dx(uv) - d/dx(integral of v * du/dx).
For d/dx(uv), we'll use the product rule: d/dx(uv) = du/dx * v + u * dv/dx. For d/dx(integral of v * du/dx), we'll just use the fact that integration is the inverse of differentiation, d/dx(integral of v * du/dx) = v * du/dx. So, d/dx(uv) - d/dx(integral of v * du/dx) = (du/dx * v + u * dv/dx) - (du/dx * v) = u * dv/dx. Since the derivative of uv - integral of v * du/dx = u * dv/dx, the integral of u * dv/dx = uv - integral of v * du/dx.