Let $f ( x , y , z ) = x y \sin z - y z \sin x$ . Then $f _ { z } ( x , y , z )$ is

A) $x y \cos z + y \sin x$
B) $x y \cos z - y \sin x$
C) $- x y \cos z - y \sin x$
D) $- x y \cos z + y \sin x$
E) $x y \sin z - y \cos x$

Question

Let $f ( x , y , z ) = x y \sin z - y z \sin x$ . Then $f _ { z } ( x , y , z )$ is

A) $x y \cos z + y \sin x$
B) $x y \cos z - y \sin x$
C) $- x y \cos z - y \sin x$
D) $- x y \cos z + y \sin x$
E) $x y \sin z - y \cos x$

Quizplus · Accepted Answer

To find $f_z(x, y, z)$, we differentiate $f(x, y, z)$ with respect to $z$, treating $x$ and $y$ as constants. The derivative of $x y \sin z$ with respect to $z$ is $x y \cos z$, and the derivative of $- y z \sin x$ with respect to $z$ is $- y \sin x$, since $\sin x$ is treated as a constant and the derivative of $z$ is $1$. Therefore, $f_z(x, y, z) = x y \cos z - y \sin x$.

Let f(x,y,z)=xysin⁡z−yzsin⁡xf ( x , y , z ) = x y \sin z - y z \sin xf(x,y,z)=xysinz−yzsinx

Let $f ( x , y , z ) = x y \sin z - y z \sin x$