Differentiating a mathematical function from first rule or "ab initio" means differentiating the method taking the definition of derivative, only without creating use of several theorems on derivative of standard forms or differentiation.
Example: Differentiate tan-1 x from first rules.
Solution: Let y = tan-1x => x = tan y
=> x + Δx = tan (y + Δy)
Let Δx ->0 so that Δy ->0. Then
