Step by step
The question
Question: find the gradient of f(x, y) = x² + y³ at (2, 1).
1 The partial derivatives
Term by term.
x²: the power 2 comes down.
The power goes down by 1.
So 2x.
y³ 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.
y³: the power 3 comes down.
The power goes down by 1.
So 3y².
Drop the zeros: ∂f/∂y = 3y².
2 Find the gradient
The gradient ∇f puts the partial derivatives together as one vector.
So ∇f = ⟨2x, 3y²⟩.
At the point, put x = 2 and y = 1 in: ∇f(2, 1) = ⟨4, 3⟩.
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 5 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 5 to about 5.05.
Uphill.
Right!
Let's look back.
First: the partial derivatives.
Then: find the gradient.
Last: check.