I'm not sure how I'm supposed to approach this. I tried using the limit definition below, and then plugging in small values near 0 for h and it seemed like it would be 1.. but I think I'm way off. I know how to take partial derivatives with respect to x, y the regular way but I'm confused on how to do this with (4). I checked in my book, but it's not really straight forward. I couldn't find a similar example online either. Please help.
Best Answer
In each of these problems, the idea is to simplify the difference quotient so that the denominator does not tend to zero as $h\to 0$.
For instance, let $f(x,y) = x^2y$. This is not your problem, but once you understand this problem, you will know how to do your problems. Then
\begin{align*} f_x(x,y) &= \lim_{h\to 0} \frac{f(x+h,y) - f(x,y)}{h} \\&= \lim_{h\to 0} \frac{(x+h)^2y - x^2y}{h} \\&= \lim_{h\to 0} \frac{(x^2+2xh+h^2)y - x^2y}{h} \\&= \lim_{h\to 0} \frac{x^2y+2xhy+h^2y - x^2y}{h} \\&= \lim_{h\to 0} \frac{2xhy+h^2y}{h} \\&= \lim_{h\to 0} (2xy+hy) = 2xy \end{align*}