The function $f: \mathbb{R} \longrightarrow \mathbb{R}$ is defined by the rule $f(x,y) = \begin{cases} \frac{x^5y}{x^4+y^2}, & (x,y) \neq (0,0), \\ 0, & (x,y)=(0,0). \end{cases}$ Evaluate $f_x(0,y)$, $f_x(0,0)$, $f_y(x,0)$, and $f_y(0,0)$. (The definition of partial derivative as a limit is the recommended method.)
My question is really a conceptual question, that I am struggling to understand. At the point $(0,0)$, the function is defined to be $0$. So, finding the partial derivatives at those points, why would you not use $f(x,y)=0$?
My instincts tell me that no, you would use the other part of the function (partly because the question suggests using the limit definition): $\dfrac{x^5y}{x^4+y^2}$, $(x,y) \neq (0,0)$. But why is this?
Secondly, in general, why would you use the limit definition of a derivative, as opposed to simply taking the derivative mechanically? What is the difference?
Thanks for any help!