Let $f ( x , y ) = \left( x ^ { 3 } + y ^ { 4 } \right) ^ { 5 }$ . Find the value of $f _ { x y } - f _ { y x }$ at the point $( 1,2 )$ .
A) $- 32$
B) $- 16$
C) $- 8$
D) $- 4$
E)0
F)4
G)8
H)16

Question

Let $f ( x , y ) = \left( x ^ { 3 } + y ^ { 4 } \right) ^ { 5 }$ . Find the value of $f _ { x y } - f _ { y x }$ at the point $( 1,2 )$ .
A) $- 32$ 
B) $- 16$
C) $- 8$ 
D) $- 4$
E)0
F)4
G)8
H)16

Quizplus · Accepted Answer

0The mixed partial derivatives $f_{xy}$ and $f_{yx}$ of a function $f(x, y)$ are equal if the function is continuous and differentiable in the region of interest, according to Clairaut's theorem. Therefore, $f_{xy} - f_{yx} = 0$ at any point, including $(1,2)$.

Let f(x,y)=(x3+y4)5f ( x , y ) = \left( x ^ { 3 } + y ^ { 4 } \right) ^ { 5 }f(x,y)=(x3+y4)5

Let $f ( x , y ) = \left( x ^ { 3 } + y ^ { 4 } \right) ^ { 5 }$