Step by step
The question
Question: find the directional derivative of f(x, y) = 3x² − y at (1, 2), in the direction of ⟨-3, 4⟩.
1 The partial derivatives
Term by term.
3x²: the power 2 comes down: 3 times 2 is 6.
The power goes down by 1.
So 6x.
y has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂x = 6x.
Term by term.
3x² has no y: it is a fixed number, so it gives 0.
y gives 1.
Drop the zeros: ∂f/∂y = −1.
2 Find the gradient
The gradient ∇f puts the partial derivatives together as one vector.
So ∇f = ⟨6x, −1⟩.
At the point, put x = 1 and y = 2 in: ∇f(1, 2) = ⟨6, −1⟩.
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.
6 times −3/5 is −18/5.
−1 times 4/5 is −4/5.
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.