This chapter interestingly discusses the existence of “differentiable” fractal interpolation functions, in contrary to the above discussed non-differentiable fractal functions, and their fractional calculus. Barnsley has discussed the calculus of fractal interpolation functions in Ref. [1] and he has demonstrated that the FIFs can be indefinitely integrated such that the resultant functions are again attractors for a new IFS. If f is the fractal interpolation function associated with the IFS \(\{(L_n(x),F_n(x,y)):n=1,2,\ldots ,N-1\}\) and \( \hat{f}(x)=\hat{y}_1+\int _{x_1}^x f(t)dt,\) then \(\hat{f}\) is also a FIF generated by \(\{(L_n(x),\hat{F}_n(x,\hat{y})):n=1,2,\ldots ,N-1\}\) , where \(\hat{F}_n(x,\hat{y})=\alpha _na_n\hat{y}+\hat{q}_n(x)\) (refer Theorem 1 in [1]).