y = log_2 [log_3 (log_5 x)] y� = ?


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Question Number 63888 by Mikael last updated on 10/Jul/19

$y = {l o g}_{2} [{l o g}_{3} ({l o g}_{5} x)]$ $y = ?$
	Answered by Hope last updated on 10/Jul/19

	$y = {l o g}_{2} [{l o g}_{3} (u)] \to u = {l o g}_{5} x$ $x = 5^{u} \to \frac{d x}{d u} = 5^{u} l n 5 = x l n 5$ $y = {l o g}_{2} [v] \to v = {l o g}_{3} u$ $3^{v} = u$ $\frac{d u}{d v} = 3^{v} l n 3 = u l n 3 = ({l o g}_{5} x) l n 3$ $y = {l o g}_{2} v$ $v = 2^{y}$ $\frac{d v}{d y} = 2^{y} l n 2 = v l n 2 = 2^{y} l n 2$ $n o w \frac{d v}{d y} \times \frac{d u}{d v} \times \frac{d x}{d u} = (2^{y} l n 2) \times ({l o g}_{5} x l n 3) \times (x l n 5)$ $\frac{d x}{d y} = (l n 2 \times l n 3 \times l n 5) ({x l o g}_{5} x) \times (2^{y})$ $\frac{d x}{d y} = (l n 2 \times l n 3 \times l n 5) \times ({x l n}_{5} x) \times 2^{{l o g}_{2} ({l o g}_{3} ({l o g}_{5} x))}$ $\frac{d y}{d x} = \frac{1}{(l n 2 \times l n 3 \times l n 5)} \times \frac{1}{{x l n}_{5} x} \times \frac{1}{2^{{l o g}_{2} ({l o g}_{3} ({l o g}_{5} x))}}$ $p l s c h e c k$
	Commented by Mikael last updated on 10/Jul/19

	$t h a n k y o u S i r$
	Answered by mr W last updated on 10/Jul/19

	$y^{'} = \frac{1}{\ln 2 \times \log_{3} (\log_{5} x)} \times \frac{1}{\ln 3 \times \log_{5} x} \times \frac{1}{\ln 5 \times x}$ $y^{'} = \frac{1}{\ln 2 \times \ln 3 \times \ln 5 \times [\log_{3} (\log_{5} x)] [\log_{5} x] x}$
	Commented by Mikael last updated on 10/Jul/19

	$T h a n k y o u S i r .$
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