Find the partial derivatives

Question

Let $f(x,y) = \frac{xy^2-x^2y+3x^3-y^3}{x^2+y^2}$ when $(x,y)\neq(0,0)$ and $f(x,y)=0$ when $(x,y)=(0,0)$.

How do I calculate $f_x(0,0)$ and $f_y(0,0)$ when I directly put (0,0) into $f_x$ and $f_y$, the denominators become zero?

Thanks!

Answer 1

Since $f(x,0)=3x$,\begin{align}\frac{\partial f}{\partial x}(0,0)&=\lim_{x\to0}\frac{f(x,0)-f(0,0)}x\\&=\lim_{x\to0}3\\&=3.\end{align}The computation of $\frac{\partial f}{\partial y}(0,0)$ is similar.