Final answer:
The nth derivative of f(x)=(x+1)^(-1) is found using the power rule for derivatives, resulting in f^(n)(x) = (-1)^n * n! * x^(-n-1), where the sign alternates and the factorial and power terms increase with n.
Step-by-step explanation:
To find the nth derivative of the function f(x)=(x+1)^(-1), we use the general formula for the nth derivative of a power of x, which is:
d^n/dx^n(x^m) = (m)(m-1)(m-2)...(m-n+1)x^(m-n), where n ≤ m
In this case, the function can be written as x^(-1), so the nth derivative of f(x) will be:
f(n)(x) = (-1)(-2)...(-n)x^(-n-1) = (-1)^n * n! * x^(-n-1)
Each new derivative increases the factorial term by one and decreases the power of x by one while alternating the sign. That is, the nth term of the derivative will be a product of factorial of n, a power of (-1) for alternating signs, and the power of x decreased by n+1.
Taking the derivatives of the function n number of times is known as nth derivative of the function. A general formula for all of the successive derivatives exists. This formula is called the nth derivative, f'n(x).