Why do all elementary functions have an elementary derivative? Considering many elementary functions have an antiderivative which is not elementary, why does this type of thing not also happen in differential calculus?

Elementary functions are linear combination or quotient of polynomial and exponential functions (yes, that include sin, cos, tan). No doubt polynomials has derivative to be polynomials. And exponential is the single most important function in the universe, and clearly it is always differentiable and produces only exponential functions. The whole process of taking derivative is a combination of above process, so they are certainly elementary.
