Question Number 140294 by solihin last updated on 06/May/21
Answered by Satyendra last updated on 06/May/21
$$\left.{c}\right)\:\mathrm{sin}^{−\mathrm{1}} {y}^{\mathrm{2}} =\mathrm{sin}{h}\left({xy}\right) \\ $$$$\Rightarrow\frac{\mathrm{2}{y}}{\:\sqrt{\mathrm{1}−{y}^{\mathrm{4}} }}\:{y}'=\mathrm{cos}{h}\left({xy}\right) \left({y}+{xy}'\right) \\ $$$$\Rightarrow\frac{\mathrm{2}{y}}{\:\sqrt{\mathrm{1}−{y}^{\mathrm{4}} }}\:{y}'={y}\mathrm{cos}{h}\left({xy}\right)+{xy}'\mathrm{cos}{h}\left({xy}\right) \\ $$$$\Rightarrow\frac{\mathrm{2}{y}}{\:\sqrt{\mathrm{1}−{y}^{\mathrm{4}} }}\:{y}'−{xy}'\mathrm{cosh}\:\left({xy}\right)={y}\mathrm{cosh}\:\left({xy}\right) \\ $$$$\Rightarrow{y}'\left[\frac{\mathrm{2}{y}}{\:\sqrt{\mathrm{1}−{y}^{\mathrm{4}} }}−{x}\mathrm{cosh}\:\left({xy}\right)\right]={y}\mathrm{cosh}\:\left({xy}\right) \\ $$$$\Rightarrow{y}'=\frac{{y}\mathrm{cosh}\:\left({xy}\right)}{\frac{\mathrm{2}{y}}{\:\sqrt{\mathrm{1}−{y}^{\mathrm{4}} }}−{x}\mathrm{cosh}\:\left({xy}\right)} \\ $$$$\Rightarrow{y}'=\frac{{y}\mathrm{cosh}\:\left({xy}\right)\sqrt{\mathrm{1}−{y}^{\mathrm{4}} }}{\mathrm{2}{y}−{x}\mathrm{cosh}\:\left({xh}\right)\sqrt{\mathrm{1}−{y}^{\mathrm{4}} }} \\ $$$$\Rightarrow\frac{{dy}}{{dx}}=\frac{{y}\mathrm{cosh}\:\left({xy}\right)\sqrt{\mathrm{1}−{y}^{\mathrm{4}} }}{\mathrm{2}{y}−{x}\mathrm{cosh}\:\left({xh}\right)\sqrt{\mathrm{1}−{y}^{\mathrm{4}} }} \\ $$
Answered by Satyendra last updated on 06/May/21
$$\left({a}\right)\:{y}=\left(\mathrm{cos}^{−\mathrm{1}} \mathrm{4}{x}^{\mathrm{2}} \right)^{\mathrm{5}} \\ $$$$\Rightarrow\frac{{dy}}{{dx}}=\frac{{d}}{{dx}}\left(\mathrm{cos}^{−\mathrm{1}} \mathrm{4}{x}^{\mathrm{2}} \right)^{\mathrm{5}} \\ $$$$\Rightarrow\frac{{dy}}{{dx}}=\mathrm{5} \left(\mathrm{cos}^{−\mathrm{1}} \mathrm{4}{x}^{\mathrm{2}} \right)^{\mathrm{4}} \:\:\frac{{d}}{{dx}\:}\left(\mathrm{cos}^{−\mathrm{1}} \mathrm{4}{x}^{\mathrm{2}} \right) \\ $$$$\Rightarrow\frac{{dy}}{{dx}}=\mathrm{5} \left(\mathrm{cos}^{−\mathrm{1}} \mathrm{4}{x}^{\mathrm{2}} \right)^{\mathrm{4}} \:\frac{\mathrm{1}}{\:−\sqrt{\mathrm{1}−\left(\mathrm{4}{x}^{\mathrm{2}} \right)^{\mathrm{2}} }}\:\frac{{d}}{{dx}}\left(\mathrm{4}{x}^{\mathrm{2}} \right) \\ $$$$\Rightarrow\frac{{dy}}{{dx}}=\mathrm{5} \left(\mathrm{cos}^{−\mathrm{1}} \mathrm{4}{x}^{\mathrm{2}} \right)^{\mathrm{4}} \:\frac{\mathrm{1}}{−\sqrt{\mathrm{1}−\mathrm{16}{x}^{\mathrm{4}\:} }}\: \mathrm{8}{x} \\ $$$$\Rightarrow\frac{{dy}}{{dx}}=−\frac{\mathrm{40}{x}\left(\mathrm{cos}^{−\mathrm{1}} \mathrm{4}{x}^{\mathrm{2}} \right)^{\mathrm{4}} }{\:\sqrt{\mathrm{1}−\mathrm{16}{x}^{\mathrm{4}} }} \\ $$
Answered by liberty last updated on 06/May/21
$$\left(\mathrm{a}\right)\:\mathrm{y}=\left(\mathrm{cos}^{−\mathrm{1}} \mathrm{4x}^{\mathrm{2}} \right)^{\mathrm{5}} \\ $$$$\Rightarrow\mathrm{y}^{\frac{\mathrm{1}}{\mathrm{5}}} \:=\:\mathrm{cos}^{−\mathrm{1}} \left(\mathrm{4x}^{\mathrm{2}} \right) \\ $$$$\Rightarrow\:\mathrm{4x}^{\mathrm{2}} \:=\:\mathrm{cos}\:\left(\mathrm{y}^{\frac{\mathrm{1}}{\mathrm{5}}} \right) \\ $$$$\Rightarrow\mathrm{8x}\:=\:\frac{\mathrm{1}}{\mathrm{5}}\mathrm{y}^{−\frac{\mathrm{4}}{\mathrm{5}}} \left(−\mathrm{sin}\:\mathrm{y}^{\frac{\mathrm{1}}{\mathrm{5}}} \right).\frac{\mathrm{dy}}{\mathrm{dx}} \\ $$$$\Rightarrow\frac{\mathrm{dy}}{\mathrm{dx}}\:=−\:\frac{\mathrm{40x}}{\mathrm{y}^{−\frac{\mathrm{4}}{\mathrm{5}}} \left(\mathrm{sin}\:\mathrm{y}^{\frac{\mathrm{1}}{\mathrm{5}}} \right)} \\ $$$$\Rightarrow\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:−\mathrm{40x}\:\mathrm{csc}\:\left(\mathrm{cos}^{−\mathrm{1}} \left(\mathrm{4x}^{\mathrm{2}} \right)\right).\left(\mathrm{cos}^{−\mathrm{1}} \left(\mathrm{4x}^{\mathrm{2}} \right)\right)^{\mathrm{4}} \\ $$
Answered by qaz last updated on 06/May/21
$$\ll{c}\gg \\ $$$$\frac{\mathrm{2}{y}}{\:\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }}{y}'=\mathrm{cosh}\:\left({xy}\right)\left[{y}+{xy}'\right]={y}\mathrm{cosh}\:\left({xy}\right)+{xy}'\mathrm{cosh}\:\left({xy}\right) \\ $$$$\Rightarrow{y}'=\frac{{y}\mathrm{cosh}\:\left({xy}\right)}{\frac{\mathrm{2}{y}}{\:\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }}−{x}\mathrm{cosh}\:\left({xy}\right)} \\ $$