Chapter 12.6, Problem 61E

Gradients in three dimensions Consider the following functions f, points P, and unit vectors u.

1. a. Compute the gradient of f and evaluate it at P
2. b. Find the unit vector in the direction of maximum increase of f at P.
3. c. Find the rate of change of the function in the direction of maximum increase at P.
4. d. Find the directional derivative at P in the direction of the given vector.

61.     $f\left(x,y,z\right)=\ln \left(1+x^2+y^2+z^2\right);P\left(1,1,-1\right);\langle \frac{2}{3^{\prime }}\frac{2}{3^{\prime }}-\frac{1}{3}\rangle$