Directional Derivative Calculator

What a directional derivative measures

Partials give rates along the coordinate axes. A directional derivative answers how fast f changes when you walk in any chosen direction. If û is a unit vector, then D_u f = ∇f · û. This page is a practical directional derivative calculator workflow: compute the gradient components with labeled steps, then finish the geometry with a short vector calculation.

How to calculate the directional derivative here

1. Enter f and compute ∂f/∂x (variable = x).
2. Change the variable to y (and z if needed) to finish ∇f.
3. Take your direction vector v and form û = v / |v|.
4. Evaluate ∇f at the given point, then compute the dot product ∇f · û.

Need only the partials without the directional story? Use the partial derivative calculator or the multivariable derivative calculator.

Worked directional examples

f = x² + y², direction ⟨1, 1⟩

∇f = ⟨2x, 2y⟩. Unit vector û = ⟨1/√2, 1/√2⟩. At (1, 0), ∇f = ⟨2, 0⟩, so D_u f = 2/√2 = √2. Use the calculator for the partials; do the normalization and dot product on paper.

f = xy, direction ⟨3, 4⟩

∇f = ⟨y, x⟩. |v| = 5 so û = ⟨3/5, 4/5⟩. At (2, 1), ∇f = ⟨1, 2⟩ and D_u f = 3/5 + 8/5 = 11/5.

f = e^(xy)

Partials need the chain rule: fₓ = y e^(xy), fᵧ = x e^(xy). Run both above, evaluate at the assigned point, then dot with û.

Maximum rate of increase

The directional derivative is largest when û points the same way as ∇f; that maximum value is |∇f|. The steepest decrease uses −∇f. Checking a homework “max rate” question usually means computing the gradient magnitude after you trust the partials from this free tool.

Input and vector tips

Keep f symbolic in the calculator; plug in the point only after you have ∇f as expressions or after reading each component. Always normalize unless your instructor uses a non-unit formula. For single-variable slopes at a point (no direction vector), see the derivative at a point calculator.

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