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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