Step by step
The question
Question: find the gradient of f(x, y) = 4x − 2y + xy at (1, 2).
1 The partial derivatives
Term by term.
4x gives 4.
2y has no x: it is a fixed number, so it gives 0.
In xy, y is fixed and waits in front.
x gives 1.
So y.
Drop the zeros: ∂f/∂x = 4 + y.
Term by term.
4x has no y: it is a fixed number, so it gives 0.
2y gives 2.
In xy, x is fixed and waits in front.
y gives 1.
So x.
Drop the zeros: ∂f/∂y = −2 + x.
2 Find the gradient
The gradient ∇f puts the partial derivatives together as one vector.
So ∇f = ⟨4 + y, −2 + x⟩.
At the point, put x = 1 and y = 2 in: ∇f(1, 2) = ⟨6, −1⟩.
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 2 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 2 to about 2.061.
Uphill.
Right!
Let's look back.
First: the partial derivatives.
Then: find the gradient.
Last: check.