$\text { For } f ( x , y ) = e ^ { y } \sin 3 x \text {, evaluate } f _ { x } \text { at the point } ( \pi , 0 ) \text {. }$
A) $f _ { x } ( \pi , 0 ) = 3$
B) $f _ { x } ( \pi , 0 ) = - 3$
C) $f _ { x } ( \pi , 0 ) = - 11$
D) $f _ { x } ( \pi , 0 ) = - 6$
E) $f _ { x } ( \pi , 0 ) = 6$

Question

$\text { For } f ( x , y ) = e ^ { y } \sin 3 x \text {, evaluate } f _ { x } \text { at the point } ( \pi , 0 ) \text {. }$ 
A) $f _ { x } ( \pi , 0 ) = 3$
B) $f _ { x } ( \pi , 0 ) = - 3$
C) $f _ { x } ( \pi , 0 ) = - 11$
D) $f _ { x } ( \pi , 0 ) = - 6$
E) $f _ { x } ( \pi , 0 ) = 6$

Quizplus · Accepted Answer

We can find the partial derivative of $f(x,y)$ with respect to $x$ by treating $y$ as a constant and differentiating $\sin 3x$ with respect to $x$:
$$f_x(x,y) = 3e^y\cos 3x$$
Therefore, $f_x(\pi,0) = 3e^0\cos 3\pi = -3$. Thus, the answer is $\boxed{	ext{(B)}}$.

For f(x,y)=eysin⁡3x, evaluate fx at the point (π,0). \text { For } f ( x , y ) = e ^ { y } \sin 3 x \text {, evaluate } f _ { x } \text { at the point } ( \pi , 0 ) \text {. } For f(x,y)=eysin3x, evaluate fx​ at the point (π,0).

$\text { For } f ( x , y ) = e ^ { y } \sin 3 x \text {, evaluate } f _ { x } \text { at the point } ( \pi , 0 ) \text {. }$