In which direction is the directional derivative of $ f(x, y) = (x^2 − y^2 )/(x^2+ y^2 )$ at $(1, 1)$ equal to zero?

Question

I tried to use the definition of directional derivative and I think I need to solve vector v which gives the address but I don't know if it's okay,

Answer 1

The directional derivative in direction $v \in \mathbb{R}^2$ at $(1, 1)$ is given by $Df(1, 1) v$, where $Df(1, 1)$ is the Jacobian matrix of $f$ i.e. the matrix of partial derivatives: $Df(1, 1) = \begin{pmatrix} \frac{\partial f}{\partial x}(1, 1) & \frac{\partial f}{\partial y}(1, 1) \end{pmatrix}$. Thus, you need to find $v = (v_1, v_2)$ s.t. $\frac{\partial f}{\partial x}(1, 1)v_1 + \frac{\partial f}{\partial y}(1, 1)v_2 = 0$. This can be done, for example, by setting $v_1 = \frac{\partial f}{\partial y}(1, 1)$ and $v_2 = -\frac{\partial f}{\partial x}(1, 1)$.

Sometimes "direction" is defined to be a vector of norm 1, so you may need to normalize $v$ for your answer to be accepted.