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**On Gradients and their use -- did I do this correctly?**

**Homework Statement**

I have a function that describes a surface. (It doesn't matter what it is, as I want to be sure I am doing the problem correctly, not get the answer to a particular one, so here's a "random" one)

f(x,y) = 25-5x

^{2}-3y

^{2}

I am starting from a point on the surface (5,1,-3). So I want to know what direction to go in where I am ascending at the maximum rate -- the greatest slope.

**The attempt at a solution**

To solve this problem I took the gradient of the function above,

[itex]\nabla f(x,y) = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}) = (-10x, -6y)[/itex]

which gives me a directional vector. I take the point I am on (5,1,-3) and ignore the z coordinate for the moment. I look at (5,1). Evaluating ∇f at (a,b) gets me (-10(5), -6(1)) = (-50, -6).

That's the direction I am wanting to go towards. So to describe the vector I should travel in, is it correct to say that I want to go along (-50-5, 1-(-6)) = (-55, 7)? I feel like I missed something here, because it seems to me the z-coordinate should be in there. But if someone could let me know that I didn't make a stupid mistake, that would be much appreciated. I've looked up a lot of examples of this kind of problem and I guess I just don't trust myself.

Thanks.