Let $f ( x , y ) = x \ln ( y ) - 4 x y + x$ . Then $f _ { y } ( x , y )$ is

A) $\frac { x ( 1 + 4 y ) } { y }$
B) $\frac { x ( 1 - 4 y ) } { y }$
C) $- \frac { x ( 1 - 4 y ) } { y }$
D) $\frac { y ( 1 - 4 y ) } { x }$
E) $\frac { y ( 1 + 4 y ) } { x }$

Question

Let $f ( x , y ) = x \ln ( y ) - 4 x y + x$ . Then $f _ { y } ( x , y )$ is

A) $\frac { x ( 1 + 4 y ) } { y }$
B) $\frac { x ( 1 - 4 y ) } { y }$
C) $- \frac { x ( 1 - 4 y ) } { y }$
D) $\frac { y ( 1 - 4 y ) } { x }$
E) $\frac { y ( 1 + 4 y ) } { x }$

Quizplus · Accepted Answer

To find $f_y(x, y)$, we differentiate $f(x, y) = x \ln(y) - 4xy + x$ with respect to $y$. The derivative of $x \ln(y)$ with respect to $y$ is $\frac{x}{y}$, the derivative of $-4xy$ with respect to $y$ is $-4x$, and the derivative of $x$ with respect to $y$ is $0$ because $x$ is treated as a constant when differentiating with respect to $y$. Adding these together gives $\frac{x}{y} - 4x = \frac{x(1 - 4y)}{y}$.

Let f(x,y)=xln⁡(y)−4xy+xf ( x , y ) = x \ln ( y ) - 4 x y + xf(x,y)=xln(y)−4xy+x

Let $f ( x , y ) = x \ln ( y ) - 4 x y + x$