Question

If f:R^2 to R defined by f(x,y)={((x^2 y^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

for (x,y)$\neq$(0,0),

f(x,y) = $\frac{x^2 y^3}{x^4+y^2}$

the directional derivative of f at A=(a,b) in the direction of U=(u₁,u₂) is

D = $\lim_{t \to o} \frac{f(A+tU)-f(A)}{t} = \lim_{t \to o} \frac{f(tu_1 , tu_2)-f(0,0)}{t}$

$\lim_{t \to o} \frac{t^5 u_1^2 u_2^3}{t^3 (t^2 u_1 ^4+ u_2 ^2)}$

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