2.3.10 Gradient and directional derivative

Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given . It is required to find at point \( A \) : a) the gradient of the function and its value. b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=2 x^{4}+8 x^{2} y^{3}, A(2 ;-1), \quad \vec{a}=\{1 ;-3\} . \]