Gradient of f(x, y, z) = xyz at (1, 2, 3)

Step by step

The question

Question: find the gradient of f(x, y, z) = xyz at (1, 2, 3).

1 The partial derivatives

Term by term.

In xyz, yz is fixed and waits in front.

x gives 1.

So yz.

So ∂f/∂x = yz.

Term by term.

In xyz, xz is fixed and waits in front.

y gives 1.

So xz.

So ∂f/∂y = xz.

Term by term.

In xyz, xy is fixed and waits in front.

z gives 1.

So xy.

So ∂f/∂z = xy.

2 Find the gradient

The gradient ∇f puts the partial derivatives together as one vector.

So ∇f = ⟨yz, xz, xy⟩.

At the point, put x = 1, y = 2 and z = 3 in: ∇f(1, 2, 3) = ⟨6, 3, 2⟩.

Careful!

The answer has two parts, here three: one for each letter.

It is an arrow, not one number.

3 Check

Let's check each part with a tiny step.

Move x by 0.001: about 6.

It matches 6.

Right!

Move y by 0.001: about 3.

It matches 3.

Right!

Move z by 0.001: about 2.

It matches 2.

Right!

Let's look back.

First: the partial derivatives.

Then: find the gradient.

Last: check.