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how to find n-th derivative of a function f(x)=[x^(n)]logx what will be n-th derivate of f(x)

how to find n-th derivative of a function f(x)=[x^(n)]logx what will be n-th derivate of f(x)

Grade:12th pass

1 Answers

Riddhish Bhalodia
askIITians Faculty 434 Points
8 years ago
you have to solve this by induction
make the hypothesis by observation that
the kth derivative of f(x) where k<n is of the form
f^{(k)}(x) = n(n-1)...(n-k+1)x^{n-k}logx + n(n-1)...(n-k)[1/n + 1/(n-1) + .... + 1/(n-k)]x^{n-k-1}

and hence we get the nthe derivative as
f^{(n)}(x) = n!logx + n![1 + 1/2 + .... + 1/(n)]

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