  This work presents an analysis of fractional derivatives and fractal
derivatives, discussing their differences and similarities. The fractal
derivative is closely connected to Haussdorff's concepts of fractional
dimension geometry. The paper distinguishes between the derivative of a
function on a fractal domain and the derivative of a fractal function, where
the image is a fractal space. Different continuous approximations for the
fractal derivative are discussed, and it is shown that the $q$-calculus
derivative is a continuous approximation of the fractal derivative of a fractal
function. A similar version can be obtained for the derivative of a function on
a fractal space. Caputo's derivative is also proportional to a continuous
approximation of the fractal derivative, and the corresponding approximation of
the derivative of a fractional function leads to a Caputo-like derivative. This
work has implications for studies of fractional differential equations,
anomalous diffusion, information and epidemic spread in fractal systems, and
fractal geometry.
