Building on our understanding of single-variable functions, we now turn our attention to functions that depend on multiple inputs. This is essential because most machine learning models, from linear regression with several features to complex neural networks, involve optimizing functions with many variables or parameters. Analyzing how these functions change as we adjust their inputs requires extending the concepts of calculus.

In this chapter, you will learn the fundamentals of multivariable calculus relevant to machine learning:

• Partial Derivatives: We'll define partial derivatives, which measure how a function changes as one specific input variable changes, keeping others constant.
• The Gradient Vector: You'll learn about the gradient, often denoted as $\nabla f$. This vector collects all the partial derivatives of a function and points in the direction where the function increases most rapidly.
• Directional Derivatives and the Hessian: We will also explore how to measure change in arbitrary directions (directional derivatives) and introduce the Hessian matrix, which organizes second-order partial derivatives and helps characterize curvature.

These concepts form the basis for understanding how to navigate and optimize high-dimensional surfaces, such as the cost functions minimized during model training. Mastering gradients is key to understanding the optimization algorithms discussed later. We'll conclude with a practical exercise using NumPy to compute gradients numerically.