Question Number 49845 by mhozhez last updated on 11/Dec/18
$${y}^{{xsiny}} +{x}^{{ysinx}} =\mathrm{1} \\ $$$${determine}\:\frac{{dy}}{{dx}} \\ $$
Commented by MJS last updated on 11/Dec/18
$$\mathrm{try}\:\mathrm{this}\:\mathrm{for}\:\mathrm{yourself}.\:\mathrm{the}\:\mathrm{method}\:\mathrm{is}\:\mathrm{simple}: \\ $$$${t}_{{i}} \left({x},{y}\right)\:\mathrm{are}\:\mathrm{terms}\:\mathrm{in}\:{x}\:\mathrm{and}\:{y} \\ $$$${t}_{\mathrm{1}} \left({x},{y}\right)+{t}_{\mathrm{2}} \left({x},{y}\right)+…=\mathrm{0}\:\mathrm{is}\:\mathrm{the}\:\mathrm{same}\:\mathrm{as} \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}{t}_{{i}} \left({x},{y}\right)=\mathrm{0} \\ $$$$\mathrm{now}\:\mathrm{derivate} \\ $$$${dx}\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{d}}{{dx}}\left[{t}_{{i}} \left({x},{y}\right)\right]+{dy}\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{d}}{{dy}}\left[{t}_{{i}} \left({x},{y}\right)\right]=\mathrm{0} \\ $$$$\Rightarrow\:\frac{{dy}}{{dx}}=−\frac{\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{d}}{{dx}}\left[{t}_{{i}} \left({x},{y}\right)\right]}{\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{d}}{{dy}}\left[{t}_{{i}} \left({x},{y}\right)\right]} \\ $$$$\mathrm{an}\:\mathrm{example} \\ $$$${ax}^{\mathrm{2}} +{bxy}+{cy}^{\mathrm{2}} +{dx}+{ey}+{f}=\mathrm{0} \\ $$$${dx}\left(\mathrm{2}{ax}+{by}+\mathrm{0}+{d}+\mathrm{0}+\mathrm{0}\right)+{dy}\left(\mathrm{0}+{bx}+\mathrm{2}{cy}+\mathrm{0}+{e}+\mathrm{0}\right)=\mathrm{0} \\ $$$${dx}\left(\mathrm{2}{ax}+{by}+{d}\right)+{dy}\left({bx}+\mathrm{2}{cy}+{e}\right)=\mathrm{0} \\ $$$$\frac{{dy}}{{dx}}=−\frac{\mathrm{2}{ax}+{by}+{d}}{{bx}+\mathrm{2}{cy}+{e}} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 11/Dec/18
$${u}={y}^{{xsiny}} \\ $$$${lnu}={xsinylny} \\ $$$$\frac{\mathrm{1}}{{u}}×\frac{{du}}{{dx}}={xsiny}×\frac{\mathrm{1}}{{y}}\frac{{dy}}{{dx}}+{xlny}×{cosy}×\frac{{dy}}{{dx}}+{sinylny} \\ $$$$\frac{{du}}{{dx}}={y}^{{xsiny}} \left[\frac{{dy}}{{dx}}\left(\frac{{xsiny}}{{y}}+{xcosylny}\right)+{sinylny}\right] \\ $$$${v}={x}^{{ysinx}} \\ $$$${lnv}={ysinxlnx} \\ $$$$\frac{\mathrm{1}}{{v}}\frac{{dv}}{{dx}}=\frac{{ysinx}}{{x}}+{ylnxcosx}+{sinxlnx}\frac{{dy}}{{dx}} \\ $$$$\frac{{dv}}{{dx}}={x}^{{ysinx}} \left(\frac{{ysinx}}{{x}}+{ylnxcosx}+{sinxlnx}\frac{{dy}}{{dx}}\right) \\ $$$${u}+{v}=\mathrm{1} \\ $$$$\frac{{du}}{{dx}}+\frac{{dv}}{{dx}}=\mathrm{0} \\ $$$${y}^{{xsiny}} \left[\frac{{dy}}{{dx}}\left(\frac{{xsiny}}{{y}}+{xcosylny}\right)+{sinylny}\right]+{x}^{{ysinx}} \left[\frac{{ysinx}}{{x}}+{ylnxcosx}+{sinxlnx}\frac{{dy}}{{dx}}\right]=\mathrm{0} \\ $$$$\frac{{dy}}{{dx}}\left[{y}^{{xsiny}} \left(\frac{{xsiny}}{{y}}+{xcosylny}\right)+\left({x}^{{ysinx}} {sinxlnx}\right)\right]=−\left[{x}^{{ysinx}} \left(\frac{{ysinx}}{{x}}+{ylnxcosx}\right)+{y}^{{xsiny}} \left({sinylny}\right)\right] \\ $$$$\frac{{dy}}{{dx}}=\frac{−\left[{x}^{{ysinx}} \left(\frac{{ysinx}}{{x}}+{ylnxcosx}\right)+{y}^{{xsiny}} \left({sinylny}\right)\right]}{\left[{y}^{{xsiny}} \left(\frac{{xsiny}}{{y}}+{xcosylny}\right)+\left({x}^{{ysinx}} {sinxlnx}\right)\right]} \\ $$$${it}\:{is}\:{the}\:{answer}… \\ $$