Directional Derivative
Compute the directional derivative of f(x,y) = ax²y² in direction (u₁, u₂) at (x₀, y₀).
Interactive Sandbox
Adjust values in real-time and observe mathematical cause-and-effect
Direct computational model evaluating Directional Derivative.
Variable Mechanics
Roles, constraints & impacts
x-coordinate
x-coordinate
y-coordinate
y-coordinate
Direction vector component u₁
Direction vector component u₁
Direction vector component u₂
Direction vector component u₂
At a Glance
Gotchas, units & real-world use
Formula Nature & Mechanics
Direct computational model evaluating Directional Derivative.
Common Mistakes & Traps
- Ensure all input measurements share consistent units before calculating.
- Check for denominator values approaching zero to prevent undefined division.
- Follow standard order of operations (PEMDAS: Parentheses, Exponents, Multiply/Divide, Add/Subtract).
Dimensional Analysis & Units
Direct evaluation matching formula variable dimensions.
Real-World & Industry Application
Widely utilized across modern calculus applications, academic analysis, and engineering estimations.
Worked Example: Directional Derivative
Given:
- x0 = 1
- y0 = 2
- u1 = 1
- u2 = 1
- f=x²y², ∂f/∂x=2xy², ∂f/∂y=2x²y
- ∇f at (1,2)=(8,4)
- û=(1/√2,1/√2)
- Dᵤf=8/√2+4/√2 ≈ 8.485
Answer: 8.4853
Frequently Asked Questions
What is the Directional Derivative formula?
Dᵤf = ∇f · û. Compute the directional derivative of f(x,y) = ax²y² in direction (u₁, u₂) at (x₀, y₀).
How do I calculate Directional Derivative?
Enter x-coordinate (x0), y-coordinate (y0), Direction vector component u₁ (u1), Direction vector component u₂ (u2) into the calculator. It applies Dᵤf = ∇f · û and shows every step of the working.
Related Calculus Calculators
Partner Spotlight
