Gradient of f(x, y) = x^2 y at (1, 2)

Step by step

The question

Question: find the gradient of f(x, y) = x²y at (1, 2).

1 The partial derivatives

Term by term.

In x²y, y is fixed and waits in front.

x² gives 2x.

So 2xy.

So ∂f/∂x = 2xy.

Term by term.

In x²y, x² is fixed and waits in front.

y gives 1.

So x².

So ∂f/∂y = x².

2 Find the gradient

The gradient ∇f puts the partial derivatives together as one vector.

So ∇f = ⟨2xy, x²⟩.

At the point, put x = 1 and y = 2 in: ∇f(1, 2) = ⟨4, 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 2.041.

Uphill.

Right!

Let's look back.

First: the partial derivatives.

Then: find the gradient.

Last: check.