## Problem on directional derivatives

Let $f(x,y) = x^2 + y^3$ and $P=(1,1)$.
Find a direction at $P$ along which $f$ is not changing.
• ## Solution

Recall that
Hence, we are looking for a unit vector $\mathbf{u}$ such that $$\nabla f(1,1) \cdot \mathbf{u} = 0$$
Hence, we are looking for a unit vector $\mathbf{u}$ such that $$\langle 2, 3 \rangle \cdot \mathbf{u} = 0$$
Recall that
We can not say $\bfu = \langle -3, 2\rangle$ because this vector does not have unit length. We are after the multiple of this vector that has unit length.
Recall that