Directional Derivative Calculator
Calculate directional derivatives, gradients, and analyze multivariable function behavior with step-by-step calculus analysis. Our vector calculus calculator supports gradient analysis, maximum rate calculations, and comprehensive multivariable studies.
Last updated: December 15, 2024
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Use x and y as variables. Supported: +, -, *, /, ^, sin, cos, tan, ln, sqrt
Calculus Analysis
Directional Derivative:
4.242641
Gradient ∇f:
(2.0000, 4.0000)
Gradient Magnitude:
4.4721
Unit Vector û:
(0.7071, 0.7071)
Direction Angle:
45.00°
Analysis:
The directional derivative is 4.242641, indicating the function is function increasing at rate 4.242641 in the given direction.
Calculation Steps:
- Function: f(x,y) = x^2 + y^2
- Point: (1, 2)
- ∇f = (2.000000, 4.000000)
- Direction: (1, 1) → û = (0.707107, 0.707107)
- Dûf = ∇f · û = 4.242641
Directional Derivative Properties:
- • Formula: Dûf = ∇f · û (dot product)
- • Maximum: |∇f| in direction of gradient
- • Minimum: -|∇f| opposite to gradient
- • Zero: Perpendicular to gradient direction
Quick Example Result
For f(x,y) = x² + y² at point (1,2) in direction (1,1):
Directional Derivative = 4.243
Function increasing at rate 4.243 in the (1,1) direction
How This Calculator Works
Our directional derivative calculator applies advanced multivariable calculus principles to analyze how functions change in specific directions. The calculator uses gradient formulasand vector operations to compute directional derivatives and provide comprehensive function analysis.
Mathematical Formulas
∇f = (∂f/∂x, ∂f/∂y)û = v/|v| = (vx, vy)/√(vx² + vy²)Dûf = ∇f · û = (∂f/∂x)(ux) + (∂f/∂y)(uy)The directional derivative formula combines the gradient (which gives the direction of steepest increase) with a unit vector in the desired direction. The dot product yields the rate of change in that specific direction.
Shows gradient vectors and directional derivatives on a 3D surface
Calculus Foundation
The directional derivative extends the concept of ordinary derivatives to multivariable functions. While a regular derivative tells us the rate of change along the x-axis, directional derivatives tell us the rate of change in any direction we choose. This is fundamental in optimization, physics, and engineering applications.
- Maximum directional derivative equals the gradient magnitude |∇f|
- Minimum directional derivative equals -|∇f| (opposite to gradient direction)
- Zero directional derivative occurs perpendicular to gradient direction
- Gradient vector points toward steepest increase of the function
Sources & References
- Calculus: Early Transcendentals - James Stewart (9th Edition)Comprehensive treatment of multivariable calculus and directional derivatives
- Mathematical Association of America - Calculus Education GuidelinesProfessional standards for teaching multivariable calculus concepts
- MIT OpenCourseWare - Multivariable CalculusEducational resources and applications of directional derivatives
Need help with other calculus calculations? Check out our gradient calculator and partial derivative calculator.
Get Custom Calculator for Your PlatformExample Analysis
Problem Setup:
- Temperature: T(x,y) = 100 - x² - y²
- Point: (2, 1) on the plate
- Direction: Moving toward (3, 2)
- Find: Rate of temperature change
Solution Steps:
- ∇T = (-2x, -2y) = (-4, -2) at (2,1)
- Direction vector: (1, 1)
- Unit vector: û = (√2/2, √2/2)
- DûT = (-4)(√2/2) + (-2)(√2/2) = -3√2
Result: Temperature decreasing at rate 4.24°C per unit distance
The negative directional derivative (-3√2 ≈ -4.24) indicates that moving in the direction (1,1) from point (2,1) causes the temperature to decrease at a rate of approximately 4.24 degrees per unit distance traveled.
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