Directional Derivative Calculator
Easily calculate the directional derivative of a multivariable function online. Understand the rate of change in a specific direction with our intuitive calculator.
Variables should be denoted as x1, x2, x3,...
Enter coordinates as comma-separated values within square brackets.
Enter vector components as comma-separated values within square brackets.
Directional Derivative:
Visualization (2D)
What is Directional Derivative?
The directional derivative measures the rate of change of a multivariable function along a specific direction at a given point. Imagine you are standing on a hill represented by the function, the directional derivative tells you how steep the slope is if you walk in a particular direction.
Mathematically, for a function \( f(x_1, x_2, ..., x_n) \), the directional derivative at a point \( P(a_1, a_2, ..., a_n) \) in the direction of a unit vector \( \mathbf{u} = \langle u_1, u_2, ..., u_n \rangle \) is given by the dot product of the gradient of \( f \) at \( P \) and the direction vector \( \mathbf{u} \).
Formula: \( D_{u\mathbf{u}}f(P) = \nabla f(P) \cdot \mathbf{u} \) Where:
- \( \nabla f(P) \) is the gradient of \( f \) at point \( P \), which is a vector of partial derivatives.
- \( \mathbf{u} \) is the unit direction vector.
- \( \cdot \) represents the dot product.
This calculator helps you compute this value by taking the function, point, and direction as inputs, making it easy to understand the function's behavior in various directions.