Question

Find the directional derivative of the function at the given point in the direction of the vector v. f(x, y, z) = xyz , (3, 3, 9), v = −1, −2, 2 duf(3, 3, 9)

Answer 1

The derivative of
`f(x,y,z)` in the direction of a vector
`\mathbf u` is

`\\abla_(\mathbf u)f=\\abla f\cdot(\mathbf u)/(\|\mathbf u\|)`

With
`f(x,y,z)=xyz`, we get

`\\abla f=(yz,xz,xy)`

and
`\mathbf u=(-1,-2,2)`,

`\|\mathbf u\|=√((-1)^2+(-2)^2+2^2)=3`

Then

`\\abla_((-1,-2,2))f(3,3,9)=(27,27,9)\cdot\frac{(-1,-2,2)}3=-21`