Question Number 162516 by CM last updated on 30/Dec/21
$${differenciate}\:{using}\:{implicit}\:{function}\:\mathrm{2}{x}+\mathrm{4}{y}+\mathrm{sin}\:{xy}=\mathrm{3} \\ $$
Commented by cortano last updated on 30/Dec/21
$$\:\frac{{d}}{{dx}}\left(\mathrm{2}{x}+\mathrm{4}{y}+\mathrm{sin}\:{xy}\right)\:=\:\frac{{d}}{{dx}}\left(\mathrm{3}\right) \\ $$$$\:\mathrm{2}+\mathrm{4}{y}'+\left({y}+{xy}'\right)\:\mathrm{cos}\:{xy}\:=\mathrm{0} \\ $$$$\:\mathrm{4}{y}'+{y}\:\mathrm{cos}\:{xy}\:+{xy}'\:\mathrm{cos}\:{xy}\:=−\mathrm{2} \\ $$$$\:\left(\mathrm{4}+{x}\:\mathrm{cos}\:{xy}\right){y}'=−\mathrm{2}−{y}\:\mathrm{cos}\:{xy} \\ $$$$\:{y}'\:=\:−\frac{\mathrm{2}+{y}\:\mathrm{cos}\:{xy}}{\mathrm{4}+{x}\:\mathrm{cos}\:{xy}}\: \\ $$