If$\ y = \log^{n}{x,\ where\log^{n}{means\log{\log{\log{\ldots(repeated\ n\ times\ ),then}}}}}$ $x\log{x\log^{2}{x\log^{3}{x\ldots\log^{n - 1}{x\log^{n}x\frac{dy}{dx}\ }\ }}}$is equal to

Question

Dr. Abubacker · Accepted Answer

(d)
We have, $y = \log^{n}x$
Where $\log^{n}{means\log{\log{\ldots(repeated\ n\ times).}}}$
On differentiating w.r.t. $x,\$we get
$\ \frac{dy}{dx} = \frac{1}{x\log x\log^{2}{x\ldots\log^{n - 1}x}}$
$	herefore$ $x\log{x\log^{2}{x\ldots\log^{n - 1}{x\log^{n}{x\frac{dy}{dx}}}}}$
$= \frac{x\log{x\log^{2}{x\ldots\log^{n - 1}{x\log^{n}x}}}}{x\log{x\log^{2}{x\ldots\log^{n - 1}x}}} = \log^{n}x$

Q. If$\ y = \log^{n}{x,\ where\log^{n}{means\log{\log{\log{\ldots(repeated\ n\ times\ ),then}}}}}$ $x\log{x\log^{2}{x\log^{3}{x\ldots\log^{n - 1}{x\log^{n}x\frac{dy}{dx}\ }\ }}}$is equal to

Explanation

Reviewed by Dr. Abubacker

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