Use the definition of partial derivatives as limits to find $f _ { x } ( x , y )$ if $f ( x , y ) = 5 x ^ { 2 } - 7 x y + 2 y ^ { 2 }$.
Select the correct answer.
A) $5 y - 7 x$
B) $10 x - 7 x y$
C) $10 x - 7 y$
D) $10 x - 7$
E) $5 x - 7 y$

Question

Quizplus · Accepted Answer

Using the definition of the partial derivative, we have:

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

Substituting in $f(x,y)$:

$$f_x(x,y)=\lim_{h	o 0}\frac{5(x+h)^2-7(x+h)y+2y^2-(5x^2-7xy+2y^2)}{h}$$

Expanding the squares and simplifying:

$$f_x(x,y)=\lim_{h	o 0}\frac{10xh+5h^2-7yh}{h}$$

Canceling out the factor of $h$:

$$f_x(x,y)=10x-7y$$

Therefore, the answer is (C) $10x-7y$.

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 $f _ { x } ( x , y )$