Chapter 14: Partial Derivatives

Partial derivatives are introduced as rates of change with respect to one variable while holding others constant, with notation including subscript and Leibniz forms. Students learn how to compute higher-order partial derivatives, including mixed partials, and explore Clairaut’s Theorem on equality of mixed partials under certain conditions. The chapter covers tangent planes and linear approximations for surfaces, showing how multivariable functions can be approximated locally by planes. The chain rule is extended to functions of multiple variables, including cases with multiple intermediate variables. Directional derivatives and the gradient vector are presented as tools for measuring the rate and direction of fastest change, with applications in optimization and physics. Implicit differentiation is revisited for functions defined implicitly in multiple variables. The chapter also introduces maximum and minimum values for functions of two variables, including the Second Partial Derivative Test and Lagrange multipliers for constrained optimization. By the end, students can analyze and optimize multivariable functions, apply partial derivatives to real-world problems, and prepare for advanced topics in vector calculus and differential equations.