Give an appropriate answer.
-$f ( x , y ) = \ln y ^ { x }$
A) $\frac { \partial f } { \partial x } = 0 ; \frac { \partial f } { \partial y } = - \frac { x } { y }$
B) $\frac { \partial f } { \partial x } = x \ln y ; \frac { \partial f } { \partial y } = - \frac { x } { y }$
C) $\frac { \partial f } { \partial x } = \ln y ; \frac { \partial f } { \partial y } = - x \ln y$
D) $\frac { \partial f } { \partial x } = \ln y ; \frac { \partial f } { \partial y } = \frac { x } { y }$

Question

Quizplus · Accepted Answer

The correct partial derivatives of $f(x, y) = \ln y^x$ are obtained by applying the chain rule and properties of logarithms. For $\frac{\partial f}{\partial x}$, since $y^x$ is treated as a constant with respect to $x$, the derivative is $\ln y$. For $\frac{\partial f}{\partial y}$, applying the chain rule gives $\frac{x}{y}$, because the derivative of $\ln y$ with respect to $y$ is $\frac{1}{y}$, and then you multiply by the inner derivative, which is $x$, resulting in $\frac{x}{y}$.

Give an Appropriate Answer f(x,y)=ln⁡yxf ( x , y ) = \ln y ^ { x }f(x,y)=lnyx

Give an Appropriate Answer $f ( x , y ) = \ln y ^ { x }$