Question

What is f_x (0,0) if

Hint: use Limit

Answer 1

`f_x(0,0)` is the partial derivative of
`f` with respect to
`x` evaluated at the origin. By definition of derivative, this is the limit

`\displaystyle f_x(0,0) = \lim_(x\to0) (f(x,0) - f(0,0))/(x-0) = \lim_(x\to0) \frac{f(x,0)}x`

We have by definition of
`f`,

`f(x,0) = (x^5)/(x^4+0^4) = (x^5)/(x^4)`

so

`\displaystyle f_x(0,0) = \lim_(x\to0) \frac{(x^5)/(x^4)}x = \lim_(x\to0) (x^5)/(x^5) = \lim_(x\to0) 1 = \boxed{1}`