On the derivative of recursive functions

After I finished high school, I found an interesting math identity. I don’t know if this result is known, but here it is.

If we consider a bijective function, we can compose the function with itself, leading to recursive functions. We can derive a composed function by applying the chain rule.

Let’s define a recursive function f as a bijective function g composed on itself n times. If we take the derivative of the function f, we get the derivative of g to the power n.

The demonstration can be made using induction.


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