Directional derivative and gradient vector

[kasse]

Homework Statement

Find the directional derivative of f=sqrt(xyz) at P(2,-1,-2) in the direction of v=i+2j-2k

The Attempt at a Solution

I calculate the gradient vector and obtain grad(f) at P= <1/2, 1, 1/2>

Then I find the unit vector of v, which is <1/3, 2/3, -2/3>

The directional derivative is the dot product of these two: 1/6 + 2/3 -2/6, which is 1/2. However, the answer is supposed to be -1/6 according to my textbook. Where's my mistake?

 

[G01]

Your mistake lies in your calculation of $$\vec\nabla f$$

Its should be: $$\vec\nabla f = <1/2, -1, -1/2>$$

Check your work for that part. Other than that everything else looks good. I get -1/6 for the directional derivative with that gradient. Good Luck!

G01