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

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

Let the unit vector be <a,b>

$D_uf(x,y)=-2x^{-3}y^3.a+(\frac{3y^2}{x^2}+2y).b$

$D_uf(0,0)=0$