Maximum rate of change of f(x, y) = xy at (3, 4) Step by Step

Step by step

The question

Question: find the fastest rate of change of f(x, y) = xy at (3, 4), and its direction.

f(x, y) = xy

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.

∂f∂x = y

Term by term.

In xy, x is fixed and waits in front.

y gives 1.

So x.

So ∂f/∂y = x.

∂f∂y = x

2 Find the gradient

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

∇f = ⟨∂f∂x, ∂f∂y⟩

So ∇f = ⟨y, x⟩.

∇f = ⟨y, x⟩

At the point, put x = 3 and y = 4 in: ∇f(3, 4) = ⟨4, 3⟩.

∇f(3, 4) = ⟨4, 3⟩

3 Find the length

The fastest growth is the length of the gradient.

Its direction is the gradient itself.

The length ‖∇f‖ is Pythagoras with two parts.

Square each part, add, then take the square root.

‖∇f‖ = √4² + 3²

4 squared is 16.

3 squared is 9.

‖∇f‖ = √16 + 9

Add: 16 plus 9 is 25.

‖∇f‖ = √25

5 times 5 is 25.

5 × 5 = 25

So ‖∇f‖ = 5.

‖∇f‖ = 5

4 Check

Let's look at it on the xy grid.

The blue curve is a level curve: f is 12 all along it.

The gradient points straight uphill, at a square corner to the level curve.

A tiny step of 0.001 along the arrow: f changes about 5 per step.

That is 5.

Right!

along ∇f: f(…) − f(3, 4)0.001 ≈ 5

Let's look back.

First: the partial derivatives.

Then: find the gradient.

Then: find the length.

Last: check.

Answer