The fractal space is often associated with natural phenomena with many length scales and the functions defined on this space are usually not differentiable. First we define a $sigma$-multifractal from $sigma$-iterated function systems with probability. We introduce the measure derivative through the invariant measure of the $sigma$-multifractal. We show that the non-differentiable function on the $sigma$-multifractal can be differentiable with respect to this measure derivative. We apply this result to some examples of ordinary differential equations and diffusion processes on $sigma$-multifractal spaces.