Question

if z = xy ln(xy), then

• A. $x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=0$
• B. $y\frac{\partial z}{\partial x}=x\frac{\partial z}{\partial y}$
• C. $x\frac{\partial z}{\partial x}=y\frac{\partial z}{\partial y}$
• D. $y\frac{\partial z}{\partial x}+x\frac{\partial z}{\partial y}=0$

Answer 1

The correct option is C $x\frac{\partial z}{\partial x}=y\frac{\partial z}{\partial y}$

z = xy ln(xy)

$\frac{\partial z}{\partial x}=xy\left(\frac{y}{xy}\right)+y\ln \left(xy\right)$

$=y(1+\ln xy)$

Similarly,

$\frac{\partial z}{\partial y}=x(1+\ln xy)$

$x\frac{\partial z}{\partial x}=xy(1+\ln xy)$

$y\frac{\partial z}{\partial y}=xy(1+\ln xy)$

$x\frac{\partial z}{\partial x}=y\frac{\partial z}{\partial y}$