Most machine learning models operate on data with multiple features or involve adjusting numerous parameters. Unlike the single-variable functions $f(x)$ we've examined so far, practical models often look more like $f(x_1, x_2, \dots, x_n)$. When a function depends on several inputs, we need a way to understand how it changes when only one of those inputs varies.

This chapter introduces the technique for doing just that: partial derivatives. You'll learn:

• How to conceptualize and work with functions of multiple variables.
• The definition and calculation of partial derivatives, using notation such as $\frac{\partial f}{\partial x}$.
• How to combine partial derivatives into the gradient vector ($\nabla f$).
• The geometric significance of the gradient and its role in indicating the direction of steepest change in a function.

These concepts extend our understanding of derivatives to the multi-dimensional situations typical in machine learning optimization.