Step by step
The question
Question: find the gradient of f(x, y) = x³ − xy at (2, 1).
1 The partial derivatives
Term by term.
x³: the power 3 comes down.
The power goes down by 1.
So 3x².
In xy, y is fixed and waits in front.
x gives 1.
So y.
So ∂f/∂x = 3x² − y.
Term by term.
x³ has no y: it is a fixed number, so it gives 0.
In xy, x is fixed and waits in front.
y gives 1.
So x.
Drop the zeros: ∂f/∂y = −x.
2 Find the gradient
The gradient ∇f puts the partial derivatives together as one vector.
So ∇f = ⟨3x² − y, −x⟩.
At the point, put x = 2 and y = 1 in: ∇f(2, 1) = ⟨11, −2⟩.
Careful!
The answer has two parts: one for each letter.
It is an arrow, not one number.
3 Check
Let's look at it on the xy grid.
The blue curve is a level curve: f is 6 all along it.
The gradient points straight uphill, at a square corner to the level curve.
A tiny step of 0.01 along the arrow: f grows from 6 to about 6.112.
Uphill.
Right!
Let's look back.
First: the partial derivatives.
Then: find the gradient.
Last: check.