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How to find the nth derivative of f(x) = (x 1)^-1

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Final answer:

To find the nth derivative of f(x) = (x+1)^-1, apply the power rule and chain rule. The nth derivative is (-1)^n * n!(x+1)^-(n+1).

Step-by-step explanation:

The nth derivative of the function f(x) = (x+1)^-1 can be found using the power rule and the chain rule.

Let's find the first few derivatives to establish a pattern. The first derivative is found by using the power rule: (x+1)^-2.

The second derivative can be found by applying the power rule again: -2(x+1)^-3. The third derivative would be: 2*3(x+1)^-4.

From these derivatives, we can see that the coefficient in front of each derivative is equal to n! (the factorial of n). Therefore, the nth derivative of f(x) = (x+1)^-1 is (-1)^n * n!(x+1)^-(n+1).

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