The First Derivative Proof of tetration functions b^^x , x^^x and x^^f(x)

This paper is to find by proof the first derive of known tetration functions, fixed base iterated functions $b\uparrow \uparrow x$ , general case for $b\uparrow \uparrow f(x)$ and variable base with variable height iterated function $x\uparrow \uparrow x$. although the case of $b\uparrow \uparrow x$ is already known by using the base change method but its derive function $f(x)$ is still depend on the derive of $f(x-1)$ which gives a shortcoming derivation. However, in the coming proofs, the resulted derivative functions are proved by applying differentiation elementary concepts step by step up to the final first derive ,but an unknown limit and a non-elementary product part of the resulted derivative function still needs study, Although I included approximation method for numerical solutions.

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