Question Number 215468 by alephnull last updated on 07/Jan/25
$$\frac{\partial{x}\omega}{\partial{y}\omega}\centerdot\frac{\partial{e}}{\partial\omega}=? \\ $$$${x}={f}\left({y}\right) \\ $$$$\omega={g}\left({y}\right) \\ $$$${e}={h}\left(\omega\right) \\ $$
Answered by MrGaster last updated on 08/Jan/25
$$\frac{\partial{x}}{\partial{y}}\centerdot\frac{\partial{w}}{\partial\omega}\centerdot\frac{\partial{e}}{\partial\omega} \\ $$$$=\frac{\partial{x}}{\partial{y}}\centerdot\mathrm{1}\centerdot\frac{\partial{e}}{\partial\omega} \\ $$$$=\frac{\partial{x}}{\partial{y}}\centerdot\frac{\partial{e}}{\partial\omega} \\ $$$$\mathrm{Given}: \\ $$$${x}={f}\left({y}\right) \\ $$$$\omega={g}\left({y}\right) \\ $$$${e}={h}\left(\omega\right) \\ $$$$\mathrm{Thus}: \\ $$$$\frac{\partial{x}}{\partial{y}}={f}'\left({y}\right) \\ $$$$\frac{\partial{e}}{\partial\omega}={h}'\left(\omega\right) \\ $$$$\mathrm{Substitute}\:\mathrm{back}: \\ $$$$\frac{\partial{x}\omega}{\partial{y}\omega}\centerdot\frac{\partial{e}}{\partial\omega}={f}'\left({y}\right)\centerdot{h}'\left({g}\left({y}\right)\right) \\ $$$$\mathrm{Conclusion}: \\ $$$${f}'\left({y}\right)\centerdot{h}'\left({g}\left({y}\right)\right) \\ $$
Commented by alephnull last updated on 08/Jan/25
$$\mathrm{thank}\:\mathrm{you} \\ $$
Answered by issac last updated on 08/Jan/25
$$=\frac{{ex}}{{y}\omega} \\ $$