Directional Derivative Calculator

Compute the rate of change of f(x,y) at a point in any direction. Get gradient, unit vector, and step-by-step explanation.

Definition: D_u f(x₀,y₀) = ∇f(x₀,y₀) · û

Use x and y as variables. Supported: +, -, *, /, ^, sin, cos, tan, exp, log, sqrt.
f=x²+y², (1,1), → (1,0) sin x cos y, (0,0), ↗ (1,1) e^{xy}, (0,1), (2,-1) x³-3xy², (1,2), (3,4) xy+2y, (2,-1), ↑ (0,1) ln(x²+y²), (1,1), (1,1) √(x²+y²), (3,4), ← (-1,0)
Computing...

Understanding the Directional Derivative

The directional derivative measures the rate at which a function f(x,y) changes at a point (x₀,y₀) as we move in a specified direction v = (u,v). It generalizes partial derivatives (which are directional derivatives along the coordinate axes).

Du f(x₀,y₀) = ∇f(x₀,y₀) · û

where ∇f = (∂f/∂x, ∂f/∂y) is the gradient vector, and û = v / ||v|| is the unit vector in the direction of v.

1. Geometric interpretation

The directional derivative is the slope of the tangent line to the curve obtained by intersecting the surface z = f(x,y) with a vertical plane parallel to the direction û. It tells how steep the surface is in that direction.

2. Relationship with gradient

  • The gradient points in the direction of steepest ascent, and its magnitude is the maximum directional derivative.
  • The directional derivative is the dot product of the gradient with the unit direction vector: it is the projection of the gradient onto that direction.
  • If the direction is perpendicular to the gradient, the directional derivative is zero (direction of no change).
  • If the direction is opposite to the gradient, the directional derivative is negative (steepest descent).

3. Properties

  • Linearity: Du (af + bg) = a Du f + b Du g.
  • Product rule: Du (f g) = f Du g + g Du f.
  • Chain rule: If f is differentiable, the directional derivative exists and can be computed via the gradient.

4. Applications

  • Optimization: Finding directions of increase/decrease; gradient descent algorithms.
  • Physics: Rate of change of temperature, pressure, or potential in a given direction.
  • Machine learning: Gradient descent uses the direction opposite to the gradient to minimize a loss function.
  • Fluid dynamics: Directional derivative of velocity potential gives flow velocity in a direction.

5. Higher dimensions & differentiability

The concept extends to functions of any number of variables. For f(x₁,…,xₙ), the directional derivative in the direction of unit vector u = (u₁,…,uₙ) is ∇f · u. A function must be differentiable at the point for the gradient to exist and the directional derivative formula to hold.

How to use this calculator: Enter a function f(x,y) using standard mathematical notation (e.g., x^2 + y^2, sin(x)*cos(y), exp(x*y)). Provide the point (x₀,y₀) and the direction vector (u,v). The calculator computes the gradient analytically using nerdamer, evaluates it at the point, normalizes the direction, and outputs the directional derivative. All steps are shown.

The function must be differentiable at the point. If the gradient is undefined (e.g., division by zero), an error will appear.

Frequently Asked Questions

A zero direction vector is invalid because it does not specify a direction. The calculator will show an error. Please enter a non-zero vector.

Yes, nerdamer supports many standard functions: sin, cos, tan, exp, log (natural log), sqrt, etc. Use 'log' or 'ln' for natural log. Example: log(x) or ln(x).

If the function is not differentiable at the point (e.g., f(x,y)=|x| at x=0), the directional derivative may not exist. The calculator will attempt to compute partial derivatives symbolically and evaluate; if they are undefined, an error will be raised.

No, you can enter any non-zero vector. The calculator automatically normalizes it to obtain the unit direction.

That was a bug in an earlier version. It has been fixed by using the static method nerdamer.diff(expr, variable) for maximum compatibility. If you still encounter issues, please refresh and try again.