2.3.14 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 \( \vec{a} \). \[ z(x, y)=\tan ^{-1} \frac{y}{x}, \quad A(1 ; 3), \quad \vec{a}=\{3 ; 4\} . \]