For and any positive integer

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

For f(x)=x−1f(x)=x^{-1} and any positive integer nn, which of the following expressions equals f(n)(x)f^{(n)}(x)? (a) (−1)n n! x−n(-1)^{n}\,n!\,x^{-n} (b) n! x−(n+1)n!\,x^{-(n+1)} (c) (−1)n n! x−(n+1)(-1)^{n}\,n!\,x^{-(n+1)} (d) (−1)n−1 n! x−(n+1)(-1)^{n-1}\,n!\,x^{-(n+1)}

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