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So I just had this question asked on an exam, but I'm almost sure that it was ill-posed or impossible to solve:

Given the function $$\displaylines{ f:{{R}^2} \to {R} \cr (x,y) \mapsto {x^2} + xy + {y^2} \cr} $$

I was asked the directional derivative at the point (9, 25000) in the direction of $\vec v = (a,b,c)$ and $a,b,c \ne 0$ (I don't remember the exact values of $a, b$ and $c$ because I had to turn in the exam but I'm pretty sure it doesn't matter)

So, what I would normally do is find the gradient vector of $f$ given by $\nabla f$ and evaluate it at the given point $(9,25000)$. Then, calculating the dot product of the directional unit vector $\vec w = \alpha (a,b,c)$ such that $\left\| {\alpha (a,b,c)} \right\| = 1$.

Okay, then $\nabla {f_{(9,25000)}} \cdot \vec w$ should yield the answer.

However, $\nabla {f_{(9,25000)}}$ belongs to $ {R}^2$ while $\vec w$ belongs to ${R}^3$. Therefore, the question doesn't make sense as the direction is given by a vector in a higher-order space than the function's domain. Am I missing something?

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The only thing the I can think is if you consider the surface $$F(x,y,f(x,y))=f(x,y)-x^2-xy-y^2=0,$$ and may be the questiong was asking for the derivative of $F$ at the point $$P=(9,2500,f(9,2500)),$$ which it is zero because $F$ is constant.

With this approach at least one can calculate the derivative in the direction of a vector belonging to $\mathbb{R}^3$, but I am aware that this is a little made up.

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    I thought about the same approach but the specifically asked for the rate of change of the function in the given direction. To give more detail, they asked for the rate of change of the function in the direction orthogonal to a certain plane in R3 (I computed the cross product of two vectors in the pane and found a linear combination of the cross product such that its length was equal to 1. Also, this vector orthogonal to the given plane met the criteria above of non-zero entries.) – 2017-02-18
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    @Maximiliano Santiago, Well, in the derivative definitions that I know, the direction must be at least in the domain space of the function. Hence, if the exam question was really in a direction that does not belong to the domain, I recommend you ask about the definition of derivative used. But by the nature of the exercise, I think it is most likely to be a mistake. – 2017-02-22