Find partial derivatives, given directional derivatives.

Question

You are given that the directional derivatives of a function $f$, at the point $(a, b)$, in the direction of the two vectors $(1, 2)$ and $(−1, 1)$, are $2$ and $3$ respectively. Find the partial derivatives of $f$ at $(a, b)$.

Answer 1

We know that in general $\Bbb d f (a,b) (u,v) = \frac {\partial f} {\partial x} (a,b) u + \frac {\partial f} {\partial y} (a,b) v$. Replacing $(u,v)$ by $(1,2)$ and $(-1,1)$ gives us

$$\frac {\partial f} {\partial x} (a,b) + 2 \frac {\partial f} {\partial y} (a,b) = 2 \\ - \frac {\partial f} {\partial x} (a,b) + \frac {\partial f} {\partial y} (a,b) = 3 ,$$

which is a $2 \times 2$ linear system in the unknowns $\frac {\partial f} {\partial x} (a,b)$ and $\frac {\partial f} {\partial y} (a,b)$. Solving it gives us $\frac {\partial f} {\partial x} (a,b) = - \frac 4 3$ and $\frac {\partial f} {\partial y} (a,b) = \frac 5 3$.