Chapter 14 - Partial Derivatives - 14.6 Directional Derivatives and the Gradient Vector - 14.6 Exercises - Page 996: 4

$3+6\sqrt3$

Formula to calculate the directional derivative is: $D_uf(x,y)=f_x(x,y)m+f_y(x,y)n$ Given: $f(x,y)=xy^3-x^2$ $D_uf(x,y)=(y^3-2x) \times \cos (\pi/3)+(3xy^2) \times \sin (\pi/3)$ From the given data, we have : At $(1,2)$ $D_uf(x,y)=6 \times (\dfrac{1}{2})+12 \times (\dfrac{\sqrt 3}{2}) =3+6\sqrt3$