Directional Derivative Calculator
Work out directional derivative instantly with clear inputs, formula shown and shareable results.
About Directional Derivative Calculator
The directional derivative is the gradient dotted with a unit direction vector: D_u f = ∇f · û. It reaches its maximum |∇f| when the direction aligns with the gradient and is zero along a level curve.
Formula
Directional derivative
D_u f = ∇f · û = |∇f|·cos θ
Worked example
With the default values below, Directional Derivative Calculator returns Directional derivative of 4.94974747. Change any input above and every figure here is recalculated in your browser.
Inputs used
- ∂f/∂x
- 3
- ∂f/∂y
- 4
- Direction vector (x, y)
- 1, 1
Results
- Directional derivative
- 4.94974747
- Maximum possible rate |∇f|
- 5.00000000
- Unit direction used
- (0.707107, 0.707107)
- Fraction of the maximum rate
- 0.98994949
Step by step
- 1Values used
- 2Directional derivative
- 3Directional derivative= 4.94974747
- 4Maximum possible rate |∇f|= 5.00000000
- 5Unit direction used= (0.707107, 0.707107)
- 6Fraction of the maximum rate= 0.98994949
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
For ∇f = (3, 4) in the direction (1, 1)?
The unit direction is (0.7071068, 0.7071068), so the derivative is 7/√2 ≈ 4.9497475 — about 99% of the maximum 5.
Must the direction be normalised?
Yes, otherwise the answer scales with the arbitrary length of the vector rather than measuring a rate per unit distance.