Step by step
The question
Question: find the gradient of f(x, y, z) = x² + y² + z² at (1, 2, 2).
1 The partial derivatives
Term by term.
x²: the power 2 comes down.
The power goes down by 1.
So 2x.
y² has no x: it is a fixed number, so it gives 0.
z² has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂x = 2x.
Term by term.
x² has no y: it is a fixed number, so it gives 0.
y²: the power 2 comes down.
The power goes down by 1.
So 2y.
z² has no y: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂y = 2y.
Term by term.
x² has no z: it is a fixed number, so it gives 0.
y² has no z: it is a fixed number, so it gives 0.
z²: the power 2 comes down.
The power goes down by 1.
So 2z.
Drop the zeros: ∂f/∂z = 2z.
2 Find the gradient
The gradient ∇f puts the partial derivatives together as one vector.
So ∇f = ⟨2x, 2y, 2z⟩.
At the point, put x = 1, y = 2 and z = 2 in: ∇f(1, 2, 2) = ⟨2, 4, 4⟩.
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 2.
It matches 2.
Right!
Move y by 0.001: about 4.
It matches 4.
Right!
Move z by 0.001: about 4.
It matches 4.
Right!
Let's look back.
First: the partial derivatives.
Then: find the gradient.
Last: check.