## Problem on directional derivatives

At a particular point, the differentiable function $f(x,y)$ has directional derivatives $$D_\mathbf{\bfu_1}f = 1 \text{, and } D_\mathbf{\bfu_2}f = -2$$ where $$\mathbf{\bfu_1} = \frac{1}{\sqrt{2}} \mathbf{i} - \frac{1}{\sqrt{2}} \mathbf{j} \text{, and } \mathbf{\bfu_2} = \frac{1}{\sqrt{2}} \mathbf{i} + \frac{1}{\sqrt{2}} \mathbf{j}.$$
Find the value of $\partial_x f$ and $\partial_y f$.
• ## Solution

We are trying to find the partial derivatives of a two-variable function given information on the directional derivatives.
To do that, we try to relate the partial derivatives of $f$ to the directional derivatives of $f$.
Recall that
\begin{align}
1 &= (\partial_x f, \partial_y f)\cdot \left(\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}} \right),\\
-2 &=(\partial_x f, \partial_y f)\cdot \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right).
\end{align}
Adding and subtracting these equations gives $$\partial_x f = - \frac{1}{\sqrt{2}}, \qquad \partial_y f = -\frac{3}{\sqrt{2}}.$$