Multivariable Calculus 12 | Second Order Partial Derivatives

Higher order partial derivatives matter because they extend the familiar “first derivative test” for local extrema into multivariable settings—where the gradient replaces the single-variable derivative, and second derivatives become the missing ingredient for a sufficient criterion. In one dimension, a differentiable function has a necessary condition for a local extremum: the first derivative must vanish. But that condition alone isn’t enough; adding the second derivative yields a decision rule: a negative second derivative at the point signals a local maximum, while a positive second derivative signals a local minimum. Multivariable calculus follows the same logic in spirit, but the second-derivative piece is harder because there are many second derivatives, not just one.

For functions f: R^n → R, the first-order condition translates cleanly: the gradient must be zero at a candidate point for a local extremum. The second-order condition, however, requires defining and working with second partial derivatives—derivatives of partial derivatives. The transcript builds this definition directly from limits: for example, ∂^2f/∂x2∂x1 is obtained by first taking the partial derivative of f with respect to x2, producing a new function, and then differentiating that new function with respect to x1. In general, there are n^2 second order partial derivatives, including “pure” ones like ∂^2f/∂x1^2 and “mixed” ones like ∂^2f/∂x2∂x1.

A key complication is that the order of differentiation can matter. Swapping the order of mixed partial derivatives—going from ∂^2f/∂x2∂x1 to ∂^2f/∂x1∂x2—may produce different results in general. The transcript then narrows to a concrete example to show what happens when order does not change. Take f(x1, x2) = sin(x1·x2). The first partial derivatives are computed using the one-variable chain rule: differentiating with respect to x1 brings down a factor of x2 and a cosine term, while differentiating with respect to x2 brings down a factor of x1 and a cosine term.

From there, the four second order partial derivatives are formed. The pure second derivatives (with respect to x1 twice, or x2 twice) simplify to expressions involving −x2^2·sin(x1·x2) and −x1^2·sin(x1·x2). The mixed derivatives require the product rule because the first partial derivatives are products of factors (like x2 or x1) with trigonometric functions. After computing both mixed derivatives—∂^2f/∂x2∂x1 and ∂^2f/∂x1∂x2—the results match exactly for this example. That equality is presented as a consequence of a famous theorem (named later), which gives conditions under which mixed partial derivatives commute. The takeaway is practical: second-order information is essential for multivariable extremum tests, but whether mixed derivatives agree depends on the function’s regularity—something the next video is set to formalize.