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Directional Derivative Calculator

Compute the rate of change of f in a specified direction.

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D_u f = f_x cosθ + f_y sinθ
e.g. x^2+y^2
0 = +x direction

📖 Tutorial | Directional Derivative

1. Definition
The directional derivative D_u f is the rate of change along unit vector u=(cosθ,sinθ): D_u f = ∇f·u = f_x cosθ + f_y sinθ.
2. Symbols
θangle from +x
uunit direction
∇fgradient
D_u fdirectional derivative
3. How it works
  • Numerically find f_x, f_y;
  • Build u from θ;
  • Dot with the gradient.
4. Steps
  1. Enter f and point;
  2. Enter θ in radians;
  3. Click Calculate.
5. Example
Example: f=x²+y² at (1,1), θ=π/4.
Solution: f_x=f_y=2; D_u f=2√2≈2.828.
6. Pitfalls
Use radians; u must be a unit vector;
Max rate is along ∇f;
Perpendicular directions give zero.

❓ FAQ | Directional Derivative

Relation to partials?
Partials are directional derivatives along axes.
Maximum?
|∇f| along ∇f.
Why radians?
Trig uses radians; 90°≈1.5708.
Negative means?
Decreasing in that direction.
Projection?
D_u f is the projection of ∇f onto u.