Let $f ( x , y ) = \sin \left( x y + y ^ { 2 } \right)$ . Then $f _ { x x } ( x , y )$ is

A) $- x y ^ { 2 } \sin \left( x y + y ^ { 2 } \right)$
B) $y ^ { 2 } \sin \left( x y + y ^ { 2 } \right)$
C) $- y ^ { 2 } \sin \left( x y + y ^ { 2 } \right)$
D) $- x ^ { 2 } \sin \left( x y + y ^ { 2 } \right)$
E) $x ^ { 2 } \sin \left( x y + y ^ { 2 } \right)$

Question

Let $f ( x , y ) = \sin \left( x y + y ^ { 2 } \right)$ . Then $f _ { x x } ( x , y )$ is

A) $- x y ^ { 2 } \sin \left( x y + y ^ { 2 } \right)$
B) $y ^ { 2 } \sin \left( x y + y ^ { 2 } \right)$
C) $- y ^ { 2 } \sin \left( x y + y ^ { 2 } \right)$
D) $- x ^ { 2 } \sin \left( x y + y ^ { 2 } \right)$
E) $x ^ { 2 } \sin \left( x y + y ^ { 2 } \right)$

Quizplus · Accepted Answer

To find $f_{xx}(x, y)$, we first differentiate $f(x, y) = \sin(xy + y^2)$ with respect to $x$, and then differentiate the result again with respect to $x$. First differentiation with respect to $x$: $f_x(x, y) = \cos(xy + y^2) \cdot (y)$.Second differentiation with respect to $x$: $f_{xx}(x, y) = -\sin(xy + y^2) \cdot y^2$.Thus, the correct answer is $- y ^ { 2 } \sin \left( x y + y ^ { 2 } \right)$.

Let f(x,y)=sin⁡(xy+y2)f ( x , y ) = \sin \left( x y + y ^ { 2 } \right)f(x,y)=sin(xy+y2)

Let $f ( x , y ) = \sin \left( x y + y ^ { 2 } \right)$