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The directional derivative of a function describes how the function changes as you move from a point in a specific direction. For a function $f(x, y)$ , the directional derivative at a point $(x_0, y_0)$ in the direction of a unit vector $u = (a, b)$ is given by:

D_uf(x_0, y_0) = ∇f(x_0, y_0) · u = f_x(x_0, y_0)a + f_y(x_0, y_0)b

Here, $∇f(x_0, y_0)$ is the gradient vector at $(x_0, y_0)$ , $f_x$ and $f_y$ are the partial derivatives, and $·$ denotes the dot product.

Example:

Suppose $f(x, y) = x^2 + y^2$ and you want the directional derivative at $(1, 2)$ in the direction of vector $v = (3, 4)$ . First, normalize $v$ to get the unit vector $u = (3/5, 4/5)$ . The gradient at $(1, 2)$ is $(2x, 2y) = (2, 4)$ . The directional derivative is:

D_uf(1, 2) = (2, 4) · (3/5, 4/5) = 2*(3/5) + 4*(4/5) = (6/5) + (16/5) = 22/5

This value tells you how fast the function increases if you move from $(1, 2)$ in the direction $(3, 4)$ .


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import numpy as np

def gradient(f, x, h=1e-6):
    n = len(x)
    grad = np.zeros(n)
    for i in range(n):
        x_forward = np.array(x)
        x_backward = np.array(x)
        x_forward[i] += h
        x_backward[i] -= h
        grad[i] = (f(*x_forward) - f(*x_backward)) / (2 * h)
    return grad

def directional_derivative(f, x0, direction):
    direction = np.array(direction)
    unit_direction = direction / np.linalg.norm(direction)
    grad = gradient(f, x0)
    return np.dot(grad, unit_direction)

# Example usage:
# f(x, y) = x^2 + y^2
f = lambda x, y: x**2 + y**2
point = [1, 2]
direction = [3, 4]
result = directional_derivative(f, point, direction)
print(f"Directional derivative at {point} in direction {direction}: {result:.2f}")

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Directional Derivatives

D_uf(x_0, y_0) = ∇f(x_0, y_0) · u = f_x(x_0, y_0)a + f_y(x_0, y_0)b