A gradient is a vector that represents the direction and rate of the steepest ascent of a function. In mathematical terms, it is the multi-variable generalization of the derivative, indicating how a function changes as its input changes. The gradient is often denoted as ∇f for a function f, and it plays a crucial role in optimization problems and physics. In practical applications, gradients are used in various fields such as machine learning, where they help in minimizing loss functions through techniques like gradient descent. Understanding gradients is essential for analyzing how changes in variables affect outcomes, making them a fundamental concept in both theoretical and applied mathematics.