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Question Number 4915 by Yozzii last updated on 21/Mar/16
$${ln}\left(\frac{{d}\left\{{y}\left({x}\right)\right\}}{{dx}}\right)=\frac{{d}}{{dx}}\left({ln}\left\{{y}\left({x}\right)\right\}\right) \\ $$$${y}\left({x}\right)=? \\ $$
Commented by Yozzii last updated on 21/Mar/16
$${y}^{'} =\frac{{d}\left\{{y}\left({x}\right)\right\}}{{dx}} \\ $$$$\Rightarrow{lny}^{'} =\frac{{d}}{{dx}}\left({ln}\left\{{y}\left({x}\right)\right\}\right)=\frac{{y}^{'} }{{y}\left({x}\right)} \\ $$$${y}\left({x}\right){lny}^{'} ={y}^{'} \\ $$$$\left({y}^{'} \right)^{{y}\left({x}\right)} ={e}^{{y}^{'} } \\ $$$$\left({y}^{'} \right)^{\frac{{y}\left({x}\right)}{{y}^{'} }} ={e} \\ $$$$ \\ $$
Commented by 123456 last updated on 21/Mar/16
$${y}\:\mathrm{ln}\:{y}'={y}' \\ $$$${c}−{x}=\frac{\mathrm{1}}{{W}\left(−\frac{\mathrm{1}}{{y}}\right)}−\mathrm{ln}\:{W}\:\left(−\frac{\mathrm{1}}{{y}}\right) \\ $$$$\mathrm{by}:\mathrm{wolf}\::{v} \\ $$
Commented by Yozzii last updated on 21/Mar/16
$${I}\:{found}\:{that}\:{answer}\:{from}\:{wolfphram}\alpha \\ $$$${but}\:{I}\:{still}\:{didn}'{t}\:{understand}\:{its}\:{derivation}. \\ $$$${How}\:{is}\:{the}\:{Lambert}\:{W}\:{function}\:{used}? \\ $$$${This}\:{math}\:{area}\:{is}\:{alien}\:{to}\:{me}.\:{I}\:{just} \\ $$$${created}\:{this}\:{question}\:{for}\:{fun}\:{to}\:{see} \\ $$$${what}\:{such}\:{function}\:{y}\:{exists}. \\ $$
Commented by prakash jain last updated on 22/Mar/16
$$\mathrm{I}\:\mathrm{remember}\:\mathrm{a}\:\mathrm{similar}\:\mathrm{question}\:\mathrm{using} \\ $$$$\mathrm{log}\:\mathrm{product}\:\mathrm{function}\:\mathrm{earlier}. \\ $$