Use the definition of partial derivatives as limits to find $f _ { x } ( x , y )$ if $f ( x , y ) = 5 x ^ { 2 } - 9 x y + 2 y ^ { 2 }$ .
A) $5 y - 9 x$
B) $10 x - 9 y$
C) $10 x - 9$
D) $10 x - 9 x y$
E) $5 x - 9 y$

Question

Quizplus · Accepted Answer

To find $f_x$, we treat $y$ as a constant and take the partial derivative with respect to $x$, using the definition of partial derivatives as limits:

$$f_x(x,y) = \lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}$$

Substituting in the given function:

\begin{align*}
f_x(x,y) &= \lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}\
&= \lim_{h\to 0}\frac{(5(x+h)^2 - 9(x+h)y + 2y^2) - (5x^2 - 9xy + 2y^2)}{h}\
&= \lim_{h\to 0}\frac{5x^2 + 10xh + 5h^2 - 9xy - 9yh - 2y^2 - 5x^2 + 9xy -2y^2}{h} \
&= \lim_{h\to 0}\frac{10xh + 5h^2 - 9yh - 2y^2}{h} \
&= \lim_{h\to 0}(10x + 5h - 9y)\quad \text{(canceling }h\text{ in the numerator)}\
&= 10x - 9y
\end{align*}

Therefore, the correct answer is B.

Use the Definition of Partial Derivatives as Limits to Find fx(x,y)f _ { x } ( x , y )fx​(x,y)

Use the Definition of Partial Derivatives as Limits to Find $f _ { x } ( x , y )$