$\text { Find the limit } \lim _ { \Delta x \rightarrow 0 } \frac { f ( x + \Delta x , y ) - f ( x , y ) } { \Delta x } \text { for the function } f ( x , y ) = 4 x ^ { 2 } - 5 y$
A) $- 5 y$
B) 0
C) $4 x$
D) $5 y$
E) $8 x$

Question

$\text { Find the limit } \lim _ { \Delta x \rightarrow 0 } \frac { f ( x + \Delta x , y ) - f ( x , y ) } { \Delta x } \text { for the function } f ( x , y ) = 4 x ^ { 2 } - 5 y$ 
A) $- 5 y$
B) 0
C) $4 x$
D) $5 y$
E) $8 x$

Quizplus · Accepted Answer

This limit represents the partial derivative of the function $f(x, y)$ with respect to $x$, which calculates how $f$ changes as $x$ changes, keeping $y$ constant. The derivative of $4x^2$ with respect to $x$ is $8x$, and since $-5y$ does not depend on $x$, it disappears in the differentiation process, leaving $8x$ as the correct answer.

A) −5y- 5 y−5y B) 0 C) 4x4 x4x D) 5y5 y5y

A) $- 5 y$ B) 0
C) $4 x$ D) $5 y$