Give an appropriate answer.
-$f ( x , y ) = \left( 10 x ^ { 5 } y ^ { 4 } - 5 \right) ^ { 2 }$
A) $\frac { \partial f } { \partial x } = 50 x ^ { 4 } y ^ { 4 } ; \frac { \partial f } { \partial y } = 40 x ^ { 5 } y ^ { 3 }$
B) $\frac { \partial f } { \partial x } = 100 x ^ { 4 } y ^ { 4 } \left( 10 x ^ { 5 } y ^ { 4 } - 5 \right) ; \frac { \partial f } { \partial y } = 80 x ^ { 5 } y ^ { 3 } \left( 10 x ^ { 5 } y ^ { 4 } - 5 \right)$
C) $\frac { \partial f } { \partial x } = 80 x ^ { 5 } y ^ { 3 } \left( 10 x ^ { 5 } y ^ { 4 } - 5 \right) ; \frac { \partial f } { \partial y } = 100 x ^ { 4 } y ^ { 4 } \left( 10 x ^ { 5 } y ^ { 4 } - 5 \right)$
D) $\frac { \partial f } { \partial x } = 2 \left( 10 x ^ { 5 } y ^ { 4 } - 5 \right) ; \frac { \partial f } { \partial y } = 2 \left( 10 x ^ { 5 } y ^ { 4 } - 5 \right)$

Question

Quizplus · Accepted Answer

The correct partial derivatives are obtained by applying the chain rule to the given function $f(x, y)$. For $\frac{\partial f}{\partial x}$, the derivative of the inside function $10x^5y^4$ with respect to $x$ is $50x^4y^4$, and for $\frac{\partial f}{\partial y}$, the derivative of the inside function with respect to $y$ is $40x^5y^3$. These are then multiplied by the derivative of the outside function, which is $2(10x^5y^4-5)$, resulting in the expressions given in option B.

Give an Appropriate Answer f(x,y)=(10x5y4−5)2f ( x , y ) = \left( 10 x ^ { 5 } y ^ { 4 } - 5 \right) ^ { 2 }f(x,y)=(10x5y4−5)2

Give an Appropriate Answer $f ( x , y ) = \left( 10 x ^ { 5 } y ^ { 4 } - 5 \right) ^ { 2 }$