Find-the-n-th-derivative-of-f-x-sin-x-lnx-

Question Number 93732 by Ar Brandon last updated on 14/May/20

Find the n - th derivative of

f (x) = \sin (x) l n x

Commented by mathmax by abdo last updated on 14/May/20

f^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} {(l n x)}^{(k)} {(s i n x)}^{(n - k)}

= (l n x) s i n (x + \frac{n π}{2}) + \sum_{k = 1}^{n} C_{n}^{k} {(l n x)}^{(k)} s i n (x + \frac{(n - k) π}{2})

w e h a v e {(l n x)}^{^{'}} = \frac{1}{x} \Rightarrow {(l n (x))}^{(k)} = {(\frac{1}{x})}^{(k - 1)} = \frac{{(- 1)}^{k - 1} (k - 1)!}{x^{k}}

\Rightarrow f^{(n)} (x) = l n x s i n (x + \frac{n π}{2}) + \sum_{k = 1}^{n} C_{n}^{k} \frac{{(- 1)}^{k - 1} (k - 1)!}{x^{k}} \times s i n (x + \frac{(n - k) π}{2})