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Question Number 215468 by alephnull last updated on 07/Jan/25

((∂xω)/(∂yω))∙(∂e/∂ω)=? x=f(y) ω=g(y) e=h(ω)

$$\frac{\partial{x}\omega}{\partial{y}\omega}\centerdot\frac{\partial{e}}{\partial\omega}=? \\ $$$${x}={f}\left({y}\right) \\ $$$$\omega={g}\left({y}\right) \\ $$$${e}={h}\left(\omega\right) \\ $$

Answered by MrGaster last updated on 08/Jan/25

(∂x/∂y)∙(∂w/∂ω)∙(∂e/∂ω) =(∂x/∂y)∙1∙(∂e/∂ω) =(∂x/∂y)∙(∂e/∂ω) Given: x=f(y) ω=g(y) e=h(ω) Thus: (∂x/∂y)=f′(y) (∂e/∂ω)=h′(ω) Substitute back: ((∂xω)/(∂yω))∙(∂e/∂ω)=f′(y)∙h′(g(y)) Conclusion: f′(y)∙h′(g(y))

$$\frac{\partial{x}}{\partial{y}}\centerdot\frac{\partial{w}}{\partial\omega}\centerdot\frac{\partial{e}}{\partial\omega} \\ $$$$=\frac{\partial{x}}{\partial{y}}\centerdot\mathrm{1}\centerdot\frac{\partial{e}}{\partial\omega} \\ $$$$=\frac{\partial{x}}{\partial{y}}\centerdot\frac{\partial{e}}{\partial\omega} \\ $$$$\mathrm{Given}: \\ $$$${x}={f}\left({y}\right) \\ $$$$\omega={g}\left({y}\right) \\ $$$${e}={h}\left(\omega\right) \\ $$$$\mathrm{Thus}: \\ $$$$\frac{\partial{x}}{\partial{y}}={f}'\left({y}\right) \\ $$$$\frac{\partial{e}}{\partial\omega}={h}'\left(\omega\right) \\ $$$$\mathrm{Substitute}\:\mathrm{back}: \\ $$$$\frac{\partial{x}\omega}{\partial{y}\omega}\centerdot\frac{\partial{e}}{\partial\omega}={f}'\left({y}\right)\centerdot{h}'\left({g}\left({y}\right)\right) \\ $$$$\mathrm{Conclusion}: \\ $$$${f}'\left({y}\right)\centerdot{h}'\left({g}\left({y}\right)\right) \\ $$

Commented by alephnull last updated on 08/Jan/25

thank you

$$\mathrm{thank}\:\mathrm{you} \\ $$

Answered by issac last updated on 08/Jan/25

=((ex)/(yω))

$$=\frac{{ex}}{{y}\omega} \\ $$

Commented by MathematicalUser2357 last updated on 29/Mar/25

Did you miscalculated because: (A) you forgot the notation of derivatives (B) you forgot the first principle of derivatives (C) you thinked that as an algebra question

$$\mathrm{Did}\:\mathrm{you}\:\mathrm{miscalculated}\:\mathrm{because}: \\ $$$$\left(\mathrm{A}\right)\:\mathrm{you}\:\mathrm{forgot}\:\mathrm{the}\:\mathrm{notation}\:\mathrm{of}\:\mathrm{derivatives} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{you}\:\mathrm{forgot}\:\mathrm{the}\:\mathrm{first}\:\mathrm{principle}\:\mathrm{of}\:\mathrm{derivatives} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{you}\:\mathrm{thinked}\:\mathrm{that}\:\mathrm{as}\:\mathrm{an}\:\mathrm{algebra}\:\mathrm{question} \\ $$

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