Compute all first-order partial derivatives of the given function. $f ( x , y ) = ( 5 x + 8 y ) ^ { 3 }$
A) $f _ { x } = 15 ( 5 x + 8 y ) ^ { 2 }$ $f _ { y } = 24 ( 5 x + 8 y ) ^ { 2 }$
B) $f _ { x } = 15 ( 5 x + 8 y ) ^ { 4 }$ $f _ { y } = 24 ( 5 x + 8 y ) ^ { 4 }$
C) $f _ { x } = 15 ( x + 8 y ) ^ { 2 }$ $f _ { y } = 24 ( 5 x + 8 ) ^ { 2 }$
D) $f _ { x } = 15 ( 8 y ) ^ { 2 }$ $f _ { y } = 24 ( 5 x ) ^ { 2 }$

Question

Quizplus · Accepted Answer

To find the first-order partial derivatives of $f(x, y) = (5x + 8y)^3$, we apply the chain rule. For $f_x$, differentiate with respect to $x$, treating $y$ as a constant. This gives $3(5x + 8y)^2 \cdot 5 = 15(5x + 8y)^2$. For $f_y$, differentiate with respect to $y$, treating $x$ as a constant, yielding $3(5x + 8y)^2 \cdot 8 = 24(5x + 8y)^2$.

Compute All First-Order Partial Derivatives of the Given Function f(x,y)=(5x+8y)3f ( x , y ) = ( 5 x + 8 y ) ^ { 3 }f(x,y)=(5x+8y)3

Compute All First-Order Partial Derivatives of the Given Function $f ( x , y ) = ( 5 x + 8 y ) ^ { 3 }$