Answer to Question #115495 in Real Analysis for Sheela John

Question

If f:R^2 to R defined by f(X,y) ={x^2y^3/x^4+y^2. if (x,y)#(0,0)
0 if (X,y) =(0,0). Find the directional derivative of f(X,y) at (0,0)?

Answer 1

Direction derivative of f(X,y) at (0,0) in the direction of $\vec{u} = (u_1,u_2)$ is

$f'((0,0),(u_1,u_2)) = \lim_{t\to 0} \frac{f((0,0)+t(u_1,u_2))-f(0,0)}{t} = \lim_{t\to 0} \frac{f(tu_1,tu_2)}{t}$

$=\lim_{t\to 0} \frac{1}{t} \frac{t^5 u_1^2 u_2^3}{t^4u_1^2+t^2u_2^2}=\lim_{t\to 0} \frac{t^2 u_1^2 u_2^3}{t^2u_1^2+u_2^2}=0$