Directional derivative of f(x, y) = x^2 + y^2 at (1, 2) in the…

Step by step

The question

Question: find the directional derivative of f(x, y) = x² + y² at (1, 2), in the direction of ⟨3, 4⟩.

1 The partial derivatives

Term by term.

x²: the power 2 comes down.

The power goes down by 1.

So 2x.

y² 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.

y²: the power 2 comes down.

The power goes down by 1.

So 2y.

Drop the zeros: ∂f/∂y = 2y.

2 Find the gradient

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

So ∇f = ⟨2x, 2y⟩.

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

3 Make it a unit vector

The direction ⟨3, 4⟩ has length 5.

A direction must have length 1.

So divide each part by 5: u = ⟨3/5, 4/5⟩.

4 Dot with the gradient

The slope in direction u is the gradient dot u.

Multiply the matching parts and add.

Careful!

With ⟨3, 4⟩ itself, the answer would be 5 times too big.

2 times 3/5 is 6/5.

4 times 4/5 is 16/5.

Add: the slope is 22/5.

5 Check

Let's check: take a tiny step of 0.001 along u.

The change in f/0.001 is about 4.4.

Our answer 22/5 is 4.4.

Almost the same.

Right!

Let's look back.

First: the partial derivatives.

Then: find the gradient.

Then: make it a unit vector.

Then: dot with the gradient.

Last: check.