# Question-142788

Question Number 142788 by mohammad17 last updated on 05/Jun/21
Answered by qaz last updated on 05/Jun/21
$$\frac{\partial\mathrm{w}}{\partial\mathrm{u}}=\frac{\partial\mathrm{w}}{\partial\mathrm{x}}\centerdot\frac{\partial\mathrm{x}}{\partial\mathrm{u}}+\frac{\partial\mathrm{w}}{\partial\mathrm{y}}\centerdot\frac{\partial\mathrm{y}}{\partial\mathrm{u}}+\frac{\partial\mathrm{w}}{\partial\mathrm{z}}\centerdot\frac{\partial\mathrm{z}}{\partial\mathrm{u}} \\$$$$\:\:\:\:\:\:\:=\mathrm{3yze}^{\mathrm{xyz}} +\mathrm{3xze}^{\mathrm{xyz}} +\mathrm{3xye}^{\mathrm{xyz}} \centerdot\mathrm{2v}^{\mathrm{2}} \\$$$$\:\:\:\:\:\:\:=\mathrm{3e}^{\mathrm{xyz}} \left(\mathrm{yz}+\mathrm{xz}+\mathrm{2xyv}^{\mathrm{2}} \right) \\$$$$\:\:\:\:\:\:\:=\mathrm{3e}^{\mathrm{xyz}} \left(\mathrm{2u}\centerdot\mathrm{2uv}^{\mathrm{2}} +\mathrm{2v}^{\mathrm{2}} \left(\mathrm{u}^{\mathrm{2}} −\mathrm{4v}^{\mathrm{2}} \right)\right) \\$$$$\:\:\:\:\:\:\:=\mathrm{3e}^{\mathrm{2u}^{\mathrm{3}} \mathrm{v}^{\mathrm{2}} +\mathrm{8uv}^{\mathrm{4}} } \left(\mathrm{6u}^{\mathrm{2}} \mathrm{v}^{\mathrm{2}} −\mathrm{8v}^{\mathrm{4}} \right) \\$$$$−−−−−−−−−−−−−−−− \\$$$$\mathrm{ln}\frac{\mathrm{w}}{\mathrm{3}}=\mathrm{xyz}=\mathrm{2uv}^{\mathrm{2}} \left(\mathrm{u}^{\mathrm{2}} −\mathrm{4v}^{\mathrm{2}} \right)=\mathrm{2u}^{\mathrm{3}} \mathrm{v}^{\mathrm{2}} −\mathrm{8uv}^{\mathrm{4}} \\$$$$\frac{\mathrm{d}}{\mathrm{du}}\left(\mathrm{ln}\frac{\mathrm{w}}{\mathrm{3}}\right)=\frac{\mathrm{d}}{\mathrm{du}}\left(\mathrm{2u}^{\mathrm{3}} \mathrm{v}^{\mathrm{2}} −\mathrm{8uv}^{\mathrm{4}} \right) \\$$$$\Rightarrow\frac{\mathrm{w}'}{\mathrm{w}}=\mathrm{6u}^{\mathrm{2}} \mathrm{v}^{\mathrm{2}} −\mathrm{8v}^{\mathrm{4}} \\$$$$\Rightarrow\frac{\partial\mathrm{w}}{\partial\mathrm{u}}=\mathrm{w}\centerdot\left(\mathrm{6u}^{\mathrm{2}} \mathrm{v}^{\mathrm{2}} −\mathrm{8v}^{\mathrm{4}} \right) \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{3}\left(\mathrm{6u}^{\mathrm{2}} \mathrm{v}^{\mathrm{2}} −\mathrm{8v}^{\mathrm{4}} \right)\centerdot\mathrm{e}^{\mathrm{2u}^{\mathrm{3}} \mathrm{v}^{\mathrm{2}} +\mathrm{8uv}^{\mathrm{4}} } \\$$