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| Math Central | Quandaries & Queries |
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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 |
Shambodeb,
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.
Harley
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