Given the function f:(-infinite,1]->R, f(x)=square root(1-x), calculate the n order derivative of the function if x=0. f^(n)(0)=?
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To calculate the n-th derivative of the function, we'll start by finding out the 1st derivative.
f'(x) = -(1-x)^(-1/2)/2
Now, we'll calculate the 2nd derivative:
f"(x) = -(1-x)^(-3/2)/2*2
The 3rd derivative is:
f"'(x) = -1*3*(1-x)^(-5/2)/2^3
We'll suppose that the k-th derivative of the function is:
[f(x)]^(k) = -[1*3*5*...(2k-3)*(1-x)^(1-2k)/2]/2^k
We'll calculate the n-th derivative of the function, for x = 0:
[f(0)]^(n) = -1*3*5*...*(2n-3)/2^n
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