Question

Find the directional derivatives of
f(x ,y ,z)= x²+2y²+3z² at P₀(1,1,1) in the direction of a = i+j+k.

Answer 1

$gradient\, f(x,y,z) = (2x, 4y, 6z)$

$gradient \, f(1,1,1) = (2, 4, 6)$

$e = (1/\sqrt{\smash[b]{3}}, 1/\sqrt{\smash[b]{3}}, 1/\sqrt{\smash[b]{3}})$

The directional derivative of f at P₀ in the direction of a is

$gradient\, f(1,1,1) \cdot e= 2/\sqrt 3+ 4/\sqrt 3+ 6/\sqrt 3=4\sqrt 3.$