Question

If $z=\frac{x^{1/25}-y^{1/25}}{x^{1/20}+y^{1/20}}$, then the value of

$x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}$ is

• A. $\frac{-z}{100}$
• B. $z$
• C. $\frac{z}{2}$
• D. $\frac{z}{4}$

Answer 1

The correct option is A $\frac{-z}{100}$

$\frac{\partial z}{\partial x}=\frac{\frac{1}{25}{x}^{\frac{-24}{25}}(x^{1/20}+y^{1/20})-\frac{1}{20}{x}^{\frac{-19}{20}}(x^{1/25}-y^{1/25})}{(x^{1/20}+y^{1/20}{)}^2}$

$\frac{x\partial z}{\partial x}=\frac{\frac{1}{25}{x}^{1/25}(x^{1/20}+y^{1/20})-\frac{1}{20}{x}^{1/20}(x^{1/25}-y^{1/25})}{(x^{1/20}+y^{1/20}{)}^2}$

$\frac{y\partial z}{\partial y}=\frac{\frac{1}{25}{y}^{1/25}(x^{1/20}+y^{1/20})-\frac{1}{20}{y}^{1/20}(x^{1/25}-y^{1/25})}{(x^{1/20}+y^{1/20}{)}^2}$

$\frac{x\partial z}{\partial x}+\frac{y\partial z}{\partial y}=(\frac{1}{25}-\frac{1}{20})z$

$=\frac{-z}{100}$