- #1

songoku

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- Homework Statement
- Please see below

- Relevant Equations
- Partial derivative

Direction derivative in the direction of unit vector u = <a, b>:

Du f(x,y) = fx (x,y) a + fy (x,y) b

My attempt:

I have proved (i), it is continuous since ##\lim_{(x,y)\rightarrow (0,0)}=f(0,0)##

I also have shown the partial derivative exists for (ii), where ##f_x=0## and ##f_y=0##

I have a problem with the directional derivative. Taking u = <a, b> , I got:

$$Du =\frac{\sqrt[3] y}{3 \sqrt[3] {x^2}}a+\frac{\sqrt[3] x}{3 \sqrt[3] {y^2}}b$$

Then how to check whether the directional derivative exists or not?

Thanks