Calc 3 directional derivative question

meadow
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The question asks:
Find the directional derivative of f (x, y, z) = z ln (x/y) at (1, 1, 2) toward the point (2, 2, 1).

What I did was find the distance between the two points to be the directional vector (i+j-k) and then I took the norm of the direction vector. so my unit vector = 1/sqrt(3) * u; then I found the gradient. From there, I found the scalar product of my unit vector and the gradient to get 0. Did I approach this problem right? Does that answer seem correct to you?
 
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Seems all right to me :smile:
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

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