Gradient & Directional Derivative

Gradient vector and directional derivative of a surface

f(x, y) = x^2 + y^2 Dᵤf = ∇f · û

Gradient ∇f

(2, 4)

Magnitude |∇f|

4.472135955

The greatest rate of increase — the slope in the steepest-ascent direction.

Directional derivative Dᵤf

Unit direction û

(1, 0)

f at the point

Steepest-ascent angle

63.43494882°

Angle between ∇f and û

63.43494882°

Dᵤf = ∇f · û = |∇f|·cos θ — the projection of the gradient onto your direction.

About this calculator

The Gradient & Directional Derivative Calculator evaluates the gradient ∇f = (∂f/∂x, ∂f/∂y) of a scalar field f(x, y) at a point, reporting the partials, the magnitude |∇f| (the steepest slope), and the direction of steepest ascent in degrees. Supply a direction vector and it normalizes it to û and computes the directional derivative D_û f = ∇f · û plus the angle between û and the gradient. Zero gradients are flagged as critical points, and zero direction vectors are reported rather than divided by.

How to use the Gradient & Directional Derivative calculator

Enter the values for your problem into the input fields.
Read the result — it updates instantly as you type.
Check the formula and the visual explanation to follow how the answer was found.
Copy the page URL to share the exact calculation.

Common examples

f = x² + y² at (1, 2) → ∇f = (2, 4), |∇f| ≈ 4.4721, steepest ascent ≈ 63.43°
Same point along direction (1, 0) → directional derivative D_û f = 2
f = xy at (2, 3) along (1, 1) → D_û f ≈ 3.5355 with |∇f| ≈ 3.6056
x² + y² at the origin → ∇f = 0: a critical point, no steepest-ascent direction

Frequently asked questions

How does the Gradient & Directional Derivative calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the Gradient & Directional Derivative calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.