## Related questions with answers

Consider the function f(x, y) = 4xy / x²+y², (x, y) ≠ (0, 0) {0, (x, y) = (0, 0) and the unit vector u = 1/√2 (i + j). Does the directional derivative of f at P(0, 0) in the direction of u exist? If f(0, 0) were defined as 2 instead of 0, would the directional derivative exist?

Solutions

VerifiedWe can use the definition of the directional derivative for this exercise.:

$D_{\bold u} f(x,y) = \lim\limits_{ t \to 0 } \dfrac{ f(x+t \cos \theta, y + t \sin \theta) - f(x,y) }{ t}$

The unit vector $\bold u = \dfrac{ 1}{\sqrt 2 }(\bold i + \bold j)$ points in the direction of:

$\theta = \arctan \dfrac{ 1}{1 } = \frac{\pi}{4}$

$D_uf(0,0)=\lim\limits_{t \to 0}\dfrac{f(tu_1,tu_2)-f(0,0)}{t}=\lim\limits_{t \to 0}\dfrac{4t^2u_1u_2}{t^2(u_1^2+u_2^2)}=4u_1u_2$ which exists for all unit vectors u (also for given u in the question).

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