Directional Derivative Calculator
Find how fast a function f(x, y) changes at a point as you move along a specified direction.
Function f(x, y)
Rate of change of f along the chosen unit direction.
- 1
Direction magnitude |u|
√(3² + 4²) = 5 - 2
Unit direction û = u ÷ |u|
(3 ÷ 5, 4 ÷ 5) = (0.6, 0.8) - 3
∂f/∂x (central-difference estimate)
2[f(x+h, y) − f(x−h, y)] ÷ (2h), h = 1×10⁻⁵. - 4
∂f/∂y (central-difference estimate)
4 - 5
Dᵤf = ∂f/∂x × û₁ + ∂f/∂y × û₂
2 × 0.6 + 4 × 0.8 = 4.4000
How does this calculator work?
The directional derivative D_u f measures how fast f(x, y) changes at a point when moving along a chosen direction. It equals the gradient ∇f = (fx, fy) dotted with the unit direction û. This tool estimates the gradient by central differences, normalizes your vector, and reports D_u f along with the partial derivatives.
Formula
How this is calculated
Pick a function f(x, y) and a point (x, y), then give a direction vector (u1, u2). The calculator first estimates the gradient ∇f = (fx, fy) using the symmetric (central) difference quotient with a small step h = 1e-5: fx ≈ [f(x+h, y) − f(x−h, y)] / (2h) and fy ≈ [f(x, y+h) − f(x, y−h)] / (2h). Central differences are used because their error is O(h²), making them far more accurate than one-sided differences.
The direction is then normalized to a unit vector û = (u1, u2) / √(u1² + u2²), since the directional derivative is defined for a direction of unit length. The result is the dot product D_u f = ∇f · û = fx·û1 + fy·û2, which gives the instantaneous rate of change of f at the point in that direction.
Units follow f: D_u f has the units of f per unit length traveled. If the direction vector is the zero vector its length is zero and the result is undefined, so that case returns no answer. The maximum possible directional derivative at a point equals the gradient magnitude |∇f| and occurs in the direction of the gradient itself; moving perpendicular to the gradient gives zero.
Frequently asked questions
The directional derivative is defined along a unit direction. Normalizing (u1, u2) by its length removes the effect of the vector’s magnitude so the result reflects only the direction, not how long the vector is.
Central differences with h = 1e-5 have error proportional to h², so for smooth functions the gradient is typically accurate to many significant digits, closely matching the exact analytic partial derivatives.
The direction of the gradient ∇f itself, where D_u f equals the gradient magnitude |∇f|. The opposite direction gives the most negative value, and directions perpendicular to ∇f give zero.
Also known as
TG we-Calculate Editorial Team. (2026). Directional Derivative Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/directional-derivative-calculator
TG we-Calculate Editorial Team. "Directional Derivative Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/directional-derivative-calculator.
TG we-Calculate Editorial Team, "Directional Derivative Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/directional-derivative-calculator
@misc{wecalculate_directional_derivative_calculator, title = {Directional Derivative Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/directional-derivative-calculator}}, year = {2026}, note = {TG we-Calculate} }
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