Directional Derivatives
Rate of change in any direction.
Key idea
Rate of change in any direction.
In depth
D_u f = ∇f · u gives the rate of change in the direction of a unit vector u. The maximum rate is |∇f| in the direction of ∇f. You must normalise u first.
Worked example
f = x² + y², at (1, 2) in direction ⟨3, 4⟩/5: ⟨2, 4⟩ · ⟨0.6, 0.8⟩ = 4.4.
Formula
Real-world uses
- Computing the rate of temperature change when walking in a chosen direction on a map
Common mistake
Using a vector that is not of unit length.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Directional Derivatives?
Rate of change in any direction. It is a topic at the expert level in Calculus.
What is the formula for Directional Derivatives?
The key formula is Dᵤf = ∇f·u.
Where is Directional Derivatives used in real life?
Real-world examples: Computing the rate of temperature change when walking in a chosen direction on a map.
What should I learn before Directional Derivatives?
Make sure you are comfortable with Gradient, the previous topic in Calculus.
What is a common mistake with Directional Derivatives?
Using a vector that is not of unit length.
How can I practise Directional Derivatives?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Directional Derivatives?
Move on to Multivariable Chain Rule next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.