Step by step
The question
Question: find the directional derivative of f(x, y) = x³ + 2y² at (1, −1), in the direction of ⟨5, 12⟩.
1 The partial derivatives
Term by term.
x³: the power 3 comes down.
The power goes down by 1.
So 3x².
2y² has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂x = 3x².
Term by term.
x³ has no y: it is a fixed number, so it gives 0.
2y²: the power 2 comes down: 2 times 2 is 4.
The power goes down by 1.
So 4y.
Drop the zeros: ∂f/∂y = 4y.
2 Find the gradient
The gradient ∇f puts the partial derivatives together as one vector.
So ∇f = ⟨3x², 4y⟩.
At the point, put x = 1 and y = −1 in: ∇f(1, −1) = ⟨3, −4⟩.
3 Make it a unit vector
The direction ⟨5, 12⟩ has length 13.
A direction must have length 1.
So divide each part by 13: u = ⟨5/13, 12/13⟩.
4 Dot with the gradient
The slope in direction u is the gradient dot u.
Multiply the matching parts and add.
Careful!
With ⟨5, 12⟩ itself, the answer would be 13 times too big.
3 times 5/13 is 15/13.
−4 times 12/13 is −48/13.
Add: the slope is −33/13.
5 Check
Let's check: take a tiny step of 0.001 along u.
The change in f/0.001 is about −2.54.
Our answer −33/13 is −2.538.
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.