Compute all first-order partial derivatives of the given function. $f ( x , y ) = ( 4 x + 2 y ) ^ { 4 }$
A) $f _ { x } = 16 ( 4 x + 2 y ) ^ { 5 }$ , $f _ { y } = 8 ( 4 x + 2 y ) ^ { 5 }$
B) $f _ { x } = 16 ( x + 2 y ) ^ { 3 }$ , $f _ { y } = 8 ( 4 x + 2 ) ^ { 3 }$
C) $f _ { x } = 16 ( 2 y ) ^ { 3 }$ , $f _ { y } = 8 ( 4 x ) ^ { 3 }$
D) $f _ { x } = 16 ( 4 x + 2 y ) ^ { 3 }$ , $f _ { y } = 8 ( 4 x + 2 y ) ^ { 3 }$

Question

Quizplus · Accepted Answer

To find the first-order partial derivatives of $f(x, y) = (4x + 2y)^4$, we apply the chain rule. For $f_x$, differentiate with respect to $x$, treating $y$ as a constant. For $f_y$, differentiate with respect to $y$, treating $x$ as a constant. The correct derivatives are $f_x = 16(4x + 2y)^3$ and $f_y = 8(4x + 2y)^3$, as the power rule and chain rule give these results.

Compute All First-Order Partial Derivatives of the Given Function f(x,y)=(4x+2y)4f ( x , y ) = ( 4 x + 2 y ) ^ { 4 }f(x,y)=(4x+2y)4

Compute All First-Order Partial Derivatives of the Given Function $f ( x , y ) = ( 4 x + 2 y ) ^ { 4 }$