# Continued differentiation

Calculus Level 4

$\large \frac{a}{2^n x^{n - 1/2}}$

Suppose we differentiate the function $$f(x) = \sqrt x$$ many times, prove that the $$n^\text{th}$$ derivative of $$f(x)$$ can be stated as the expression above.

Give the last three digits for the value for $$a$$, when $$n = 20$$.

Details and Assumptions:

As an explicit example, if $$a = - 123456$$, then submit your answer as 456; if $$a = 654321$$, then submit your answer as 321.

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