Calculus Tool
Directional Derivative Calculator
Calculate the directional derivative, gradient vector, and maximum rate of change for multivariable functions. Supports vector and angle input with step-by-step solutions.
6 Preset Functions-Gradient Vector-Max Rate of Change-Step-by-Step
Directional Derivative Calculator
4.400000
Directional Derivative D_u f
(2.000000, 4.000000)
Gradient Vector
4.472136
|Gradient| (Max Rate)
Step-by-step
1. f(x,y) = x^2 + y^2
2. Gradient at (1, 2): (2.000000, 4.000000)
3. Unit vector u = (0.600000, 0.800000)
4. D_u f = grad f . u = 2.000000 x 0.600000 + 4.000000 x 0.800000 = 4.400000
Maximum increase direction: (0.447214, 0.894427) with rate 4.472136
Directional Derivative Properties
| Direction | D_u f Value | Interpretation |
|---|---|---|
| Along gradient | |grad(f)| (maximum) | Steepest uphill direction |
| Opposite gradient | -|grad(f)| (minimum) | Steepest downhill direction |
| Perpendicular to gradient | 0 | Moving along a level curve |
| Along x-axis (1,0) | df/dx (partial derivative) | Rate of change in x only |
| Along y-axis (0,1) | df/dy (partial derivative) | Rate of change in y only |
| At angle theta | fx*cos(theta) + fy*sin(theta) | General direction from x-axis |
Frequently Asked Questions
What is a directional derivative?⌄
The directional derivative measures the rate of change of a function in any specified direction. It generalizes partial derivatives, which only measure change along the x or y axes. The directional derivative in direction u is D_u f = grad(f) . u.
What is the formula for the directional derivative?⌄
D_u f(a,b) = fx(a,b) * ux + fy(a,b) * uy, where fx and fy are partial derivatives at point (a,b), and u = (ux, uy) is the unit vector in the desired direction.
What is the gradient vector?⌄
The gradient is the vector of partial derivatives: grad(f) = (df/dx, df/dy). It points in the direction of maximum increase, and its magnitude equals the maximum rate of change at that point.
In which direction is the maximum rate of change?⌄
The direction of the gradient vector. The maximum rate of change equals the magnitude of the gradient, |grad(f)|. The minimum rate of change is in the opposite direction (-grad(f)), with value -|grad(f)|.
What does it mean when the directional derivative is zero?⌄
The function has no change in that direction - you are moving along a level curve (contour line). This happens when the direction is perpendicular to the gradient vector.
How do I find the unit vector from a direction vector?⌄
Divide by the magnitude: u = v / |v|. For v = (3, 4): |v| = sqrt(9+16) = 5, so u = (3/5, 4/5) = (0.6, 0.8).
Can the directional derivative be negative?⌄
Yes. A negative directional derivative means the function is decreasing in that direction. Moving opposite to the gradient always gives the most negative directional derivative.
What is the directional derivative used for?⌄
Gradient descent in machine learning, terrain slope analysis in geography, fluid flow direction in engineering, heat flow analysis, and optimization problems in economics and physics.