fxy 5 4x2 3y at xy towards the origin please explain the

Solution

Directional derivative includes a unit vector along which it is to be found.

From (x,y) towards origin, vector is -x i - y j

Thus unit vector (U) is -(x i + y j)/( x^2 + y^2)

Grad (f) = doe(f)/doe(x) i + doe(f)/doe(y) j

The above are partial derivatives of f wrt x and y

=> Grad(f) = -8x i - (3 j)

Thus directinal derivative is Grad(f).(U)

= (-8x i -3j).(-(x i + y j)/( x^2 + y^2))

= 8x²/( x^2 + y^2) + 3y/(( x^2 + y^2)) (Ans)