EXAMPLE 5 If f(x, y, z) = x sin(yz), (a) find the gradient off and (b) find the directional derivative of fat (3, 4, 0) in the direction of v= i + 2j - k.
SOLUTION
(a) The gradient of fis
∇f(x, y, z) = (fx(x, y, z), fy(x, y, z), fz(x, y, z))
=___
(b) At (3, 4, 0) we have ∇f(3, 4,0)=___ The unit vector in the direction of v = i + 2j - k is
u=___
Therefore this equation gives
Du, (3, 4,0) = ∇f(3, 4,0). u
= 12k (___)
=___