For a positive integer

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

Problem

For a positive integer mm, let g(x)=xmg(x)=x^{m}. Derive a general formula for g(n)(x)g^{(n)}(x), where nn is a non-negative integer, expressing your answer in terms of mm, nn, and factorials. Clearly state the condition(s) on nn under which the derivative is nonzero.

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