Gradient of f(x, y) = xy at (2, 3)

Step by step

The question

Question: find the gradient of f(x, y) = xy at (2, 3).

1 The partial derivatives

Term by term.

In xy, y is fixed and waits in front.

x gives 1.

So y.

So ∂f/∂x = y.

Term by term.

In xy, x is fixed and waits in front.

y gives 1.

So x.

So ∂f/∂y = x.

2 Find the gradient

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

So ∇f = ⟨y, x⟩.

At the point, put x = 2 and y = 3 in: ∇f(2, 3) = ⟨3, 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 6.036.

Uphill.

Right!

Let's look back.

First: the partial derivatives.

Then: find the gradient.

Last: check.