We need to find the directional derivative of the function , $f(x,y) = x^{2}+y^{2}+xy$ at $P(1,-1)$ in the direction towards origin.
The direction towards origin form the point $(1,-1)$ is represented by a unit vector $u$ ,
Is it correct to take $u=\dfrac{-i-j}{\sqrt{2}}$ ?
Best Answer
The direction towards the origin $O$ from $P$ is represented by the vector $$\mathbf u = O - P = (-1, 1).$$
The unit vector will then be $$\mathbf{\hat u} = \frac{\mathbf u}{\|\mathbf u\|} = \left(-\frac1{\sqrt{2}}, \frac1{\sqrt{2}}\right) = \frac{-i + j}{\sqrt 2}.$$
We then have $$\frac{\mathrm df}{\mathrm d{\mathbf{\hat u}}} = \nabla f \cdot \mathbf{\hat u}.$$