Directional Derivative Calculator
Directional Derivative Calculator
Results
Theory Behind Directional Derivatives
The directional derivative measures the rate at which a function changes at a point in a given direction.
Definition
The directional derivative of a function f at point (x₀, y₀) in the direction of a unit vector u = (u₁, u₂) is:
Dᵤf(x₀, y₀) = ∇f(x₀, y₀) · u
Gradient
The gradient ∇f is a vector of partial derivatives:
∇f = (∂f/∂x, ∂f/∂y)
Unit Vector
If the direction vector v is not a unit vector, we normalize it:
u = v / ||v||
Interpretation
The directional derivative gives the slope of the function in the specified direction. The maximum directional derivative occurs in the direction of the gradient.
Example Functions
Click on an example to load it into the calculator:
Directional Derivative Calculator – Instantly Find Derivatives in Any Direction If you’re looking for a fast and accurate way to calculate directional derivatives online, our Directional Derivative Calculator is the perfect tool for you. Whether you’re a student tackling multivariable calculus, an engineer working with gradients, or a math enthusiast exploring vector calculus, this free online calculator will help you compute directional derivatives step by step.
What is a Directional Derivative?
The directional derivative of a function measures how the function changes as you move in a specific direction from a given point. In simple terms, it tells you the rate of change of the function in any chosen direction. This concept is widely used in calculus, optimization, engineering, and physics to analyze gradients and slopes in multidimensional spaces.
Mathematically, the directional derivative of a function f(x, y) at a point (x₀, y₀) in the direction of a vector u = (a, b) is calculated as:
D₍u₎f(x₀, y₀) = ∇f(x₀, y₀) · û
where:
∇f = gradient vector of the function
û = unit vector in the direction of u
· = dot product
Why Use a Directional Derivative Calculator?
Manually calculating directional derivatives involves:
Finding the gradient of the function (partial derivatives)
Evaluating at a point
Normalizing the direction vector
Performing a dot product
This process can be time-consuming and error-prone, especially for complex functions. Our Directional Derivative Calculator automates all these steps and provides instant, accurate results with just a few inputs.
Perfect for:
Students checking homework or assignments
Professionals needing quick derivative calculations
Teachers explaining step-by-step solutions
How to Use the Directional Derivative Calculator
Using the calculator is simple:
Enter your function f(x, y) (e.g., x² + y²)
Enter the point (x₀, y₀) where you want to evaluate the derivative
Enter the direction vector u (a, b)
Click Calculate
The calculator will display:
Gradient ∇f
Normalized unit vector
Final directional derivative D₍u₎f(x₀, y₀)
It also provides intermediate steps so you can learn how the calculation works!
Example Calculation
For example, to calculate the directional derivative of f(x, y) = x² + y² at point (1,1) in the direction of vector (1,1):
Enter x^2 + y^2
Enter 1, 1 for point
Enter 1, 1 for vector
Click Calculate
The calculator shows:
Gradient: (2x, 2y) = (2, 2)
Unit vector: (√2/2, √2/2)
Directional Derivative: D₍u₎f = (2,2)·(√2/2,√2/2) = 2√2
Try It Now!
Don’t waste time solving by hand—use our free Directional Derivative Calculator to get instant results and step-by-step solutions.
Click here to start calculating!
Whether you’re solving a classroom problem or an engineering equation, this online tool makes finding directional derivatives simple and accessible.
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