Partial derivatives give the rate of change due east and due north. A directional derivative gives it in any direction you like.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The formula
The directional derivative in the direction of a unit vector u is ∇f · u. It is a dot product, and nothing more.
The direction must be a unit vector
Using a longer vector inflates the answer. Divide by the magnitude first or the number is meaningless.
The largest possible value
The dot product is largest when u points along ∇f. That is why the gradient is the steepest ascent direction.
The zero case
Walking perpendicular to the gradient gives a directional derivative of zero. That direction is along the level curve.
The consequence
The gradient is always perpendicular to the level curve through the point. Steepest ascent is at right angles to no change at all.
The rate in a chosen direction
A directional derivative measures how fast f changes as you move in a specified direction. The partial derivatives are the special cases where the direction is along an axis.
It is a dot product
The directional derivative equals ∇f · u, where u is a unit vector. The direction must be normalised, and forgetting that is the standard error, giving an answer scaled by the wrong factor.
Maximum, minimum and zero
The rate is greatest along the gradient, most negative against it, and zero perpendicular to it. The dot product formula makes all three immediate from the cosine.
A concrete reading
On terrain, it is the slope you feel walking in a given compass direction. Steepest going one way, steepest downhill the other, level along the contour.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(3, 4) · length 5
- Vector b(1, 0) · length 1
Nested closed rings around one point mean a single peak or valley.
Step 3: Watch an Example
One step at a time.
Watch Ibrahim Walk Northeast
Ibrahim has ∇f = ⟨3, 4⟩ and wants the rate of change toward the direction ⟨3, 4⟩ itself.
- Step 1
He finds the magnitude of the direction vector: the square root of 9 + 16, which is 5.
Step 4: Your Turn
Practice makes it stick.
The Due East Walk
Problem 1 of 2
∇T = ⟨6, −2⟩. What is the rate of change walking in the unit direction ⟨1, 0⟩?
The Level Walk
Problem 2 of 2
∇f = ⟨0, 5⟩. What is the rate of change walking in the unit direction ⟨1, 0⟩?
Choose a Direction
1 of 8
∇f = ⟨2, 7⟩ and u = ⟨0, 1⟩. What is the directional derivative?
2 of 8
∇f = ⟨3, 4⟩. What is the largest possible directional derivative?
3 of 8
∇f = ⟨3, 4⟩. What is the smallest possible directional derivative?
4 of 8
Walking along a level curve. What is the directional derivative?
5 of 8
A direction vector ⟨6, 8⟩. What is its magnitude, needed before using it?
6 of 8
∇f = ⟨1, 1⟩ and u = ⟨0.6, 0.8⟩. What is the directional derivative, as a decimal?
7 of 8
Sort each direction by the directional derivative it produces.
Tap something to move it.
- Empty
- Empty
8 of 8
∇f = ⟨0, 0⟩. What is the directional derivative in any direction?
Step 5: Quick Check
Show what you know.
Question 1 of 2
∇f = ⟨8, 6⟩. What is the largest possible directional derivative?
Question 2 of 2
How is the gradient related to a level curve?
What You Learned
- A directional derivative is ∇f dotted with a unit vector in the chosen direction.
- It is largest along the gradient and zero perpendicular to it.
- Because zero happens along the level curve, the gradient is perpendicular to level curves.