With a direct attempt to calculate the first-order derivative of the function we find:
f'(k) = f(k - 1) + k * f'(k - 1),
which suggests that the equation would first need to be solved before it can be differentiated.

In the first place, are you sure that the function that you provided is actually differentiable? You didn't define whether k, a, and n are natural or real numbers but the notation suggests that k is a natural number which would mean that the derivative of the function by k is not defined. Please tell me the values that k, a, and n can assume as well as any boundary conditions so I can proceed.
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