Math CentralQuandaries & Queries


Question from shambodeb, a student:

This is a successive differentiation problem by Leibnitz theorem

If y = xn-1 log x ; Proof nth derivative y(n) = (n-1)!/x


I would prove this by induction.

It is straightforward to verify it is true for n = 1.

Inductive hypotheses: Suppose that for some positive integer k, if y = xk-1 log x then the kth derivative of y is y(k) = (k-1)!/x.

Let y = xk+1-1 log x = xk log x and find the (k + 1)st derivative of y.

y' = k xk-1 log x + xk × (1/x) = k xk-1 log x + xk-1

Differentiate k more times using the induction hypothesis.


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