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Question Number 162516 by CM last updated on 30/Dec/21

differenciate using implicit function 2x+4y+sin xy=3

$${differenciate}\:{using}\:{implicit}\:{function}\:\mathrm{2}{x}+\mathrm{4}{y}+\mathrm{sin}\:{xy}=\mathrm{3} \\ $$

Commented by cortano last updated on 30/Dec/21

 (d/dx)(2x+4y+sin xy) = (d/dx)(3)   2+4y′+(y+xy′) cos xy =0   4y′+y cos xy +xy′ cos xy =−2   (4+x cos xy)y′=−2−y cos xy   y′ = −((2+y cos xy)/(4+x cos xy))

$$\:\frac{{d}}{{dx}}\left(\mathrm{2}{x}+\mathrm{4}{y}+\mathrm{sin}\:{xy}\right)\:=\:\frac{{d}}{{dx}}\left(\mathrm{3}\right) \\ $$$$\:\mathrm{2}+\mathrm{4}{y}'+\left({y}+{xy}'\right)\:\mathrm{cos}\:{xy}\:=\mathrm{0} \\ $$$$\:\mathrm{4}{y}'+{y}\:\mathrm{cos}\:{xy}\:+{xy}'\:\mathrm{cos}\:{xy}\:=−\mathrm{2} \\ $$$$\:\left(\mathrm{4}+{x}\:\mathrm{cos}\:{xy}\right){y}'=−\mathrm{2}−{y}\:\mathrm{cos}\:{xy} \\ $$$$\:{y}'\:=\:−\frac{\mathrm{2}+{y}\:\mathrm{cos}\:{xy}}{\mathrm{4}+{x}\:\mathrm{cos}\:{xy}}\: \\ $$

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