Directional Derivative Calculator
Calculate the slope of f(x,y) in a specific direction vector v.
Variables: x, y. Supported: sin, cos, exp, ^, etc.
Note: Vector will be normalized to unit vector u automatically.
Results
1. Gradient ∇f at P
2. Unit Direction Vector u
3. Directional Derivative (Slope)
Slope Visualization (z = f(x,y))
Drag to RotateVisualization: The red arrow shows the direction you are moving. The slope is how steep the surface is along that arrow.
What is a Directional Derivative?
Imagine you are hiking on a hilly terrain represented by the function f(x, y). If you stand at a specific point on the map, you can choose to walk North, East, South-West, or any other direction.
Partial derivatives (∂f/∂x and ∂f/∂y) only tell you the slope if you walk strictly East (x-direction) or North (y-direction). But what if you walk Northeast? That’s where the Directional Derivative comes in. It calculates the slope of the hill in any specific direction you choose.
The Formula
Duf = ∇f · u
Where:
• ∇f is the Gradient Vector (contains simple partial derivatives).
• u is the Unit Vector pointing in your chosen direction.
• The “·” represents the Dot Product.
Step-by-Step Calculation
- Find the Gradient: Calculate ∂f/∂x and ∂f/∂y at your point. This gives you the vector ∇f.
- Normalize Direction: If your direction is given as a vector v, make it a unit vector u by dividing by its length (|v|). u = v / |v|.
- Dot Product: Multiply the x-components and y-components and add them up: (∇fx * ux) + (∇fy * uy).
Frequently Asked Questions (FAQ)
Q: What if the Directional Derivative is positive?
A: A positive value means you are walking uphill. The steeper the hill, the larger the number.
Q: What is the maximum possible Directional Derivative?
A: The maximum slope occurs when you walk in the exact direction of the gradient (steepest ascent). The value of the derivative equals the magnitude of the gradient, |∇f|.
Q: Can I use an angle θ instead of a vector?
A: Yes! If given an angle θ, your unit vector u is simply < cos(θ), sin(θ) >.