Step by step
The question
Question: find the gradient of f(x, y, z) = xy + yz² at (1, 1, 2).
1 The partial derivatives
Term by term.
In xy, y is fixed and waits in front.
x gives 1.
So y.
yz² has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂x = y.
Term by term.
In xy, x is fixed and waits in front.
y gives 1.
So x.
In yz², z² is fixed and waits in front.
y gives 1.
So z².
So ∂f/∂y = x + z².
Term by term.
xy has no z: it is a fixed number, so it gives 0.
In yz², y is fixed and waits in front.
z² gives 2z.
So 2yz.
Drop the zeros: ∂f/∂z = 2yz.
2 Find the gradient
The gradient ∇f puts the partial derivatives together as one vector.
So ∇f = ⟨y, x + z², 2yz⟩.
At the point, put x = 1, y = 1 and z = 2 in: ∇f(1, 1, 2) = ⟨1, 5, 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 1.
It matches 1.
Right!
Move y by 0.001: about 5.
It matches 5.
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.