2.3.13 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's 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 \( \vec{a} \). \[ z(x, y)=\sqrt{x^{2}+y^{2}}, \quad A(3 ; 4), \quad \vec{a}=\{4 ;-3\} . \]