Second Order Partial Derivatives- Understanding Multivariable Calculus

What Are Second Order Partial Derivatives?

First order partial derivatives tell you how a function changes when you tweak one variable at a time. Second order partial derivatives tell you how those rates of change themselves are changing. That's it. You're taking derivatives of derivatives.

If you already understand partial derivatives, this is the next logical step. If you don't, go back and learn that first—this article assumes you know what ∂f/∂x means.

The Four Second Order Partial Derivatives

For a function f(x, y), you have four second order partial derivatives, not two. Here's why:

• You can differentiate ∂f/∂x with respect to x again → f_xx
• You can differentiate ∂f/∂x with respect to y → f_xy
• You can differentiate ∂f/∂y with respect to x → f_yx
• You can differentiate ∂f/∂y with respect to y again → f_yy

The subscripts tell you the order. f_xy means differentiate first with respect to x, then with respect to y. This is called a mixed partial derivative.

Notation Systems

You'll encounter two main notation systems. Both are fine. Pick one and stick with it.

The Leibniz notation looks messy but shows exactly which variable you're differentiating with respect to in each step. The subscript notation is compact and what you'll see in textbooks when discussing the Hessian matrix.

Clairaut's Theorem (Equality of Mixed Partials)

Here's something that trips people up: f_xy does not always equal f_yx. They're equal under specific conditions.

Clairaut's Theorem states: If the mixed partials are continuous at a point, then they're equal there. That's the key phrase—continuous at a point.

In practice, if you're working with "nice" functions (polynomials, exponentials, sine/cosine), you can assume f_xy = f_yx. The functions you'll encounter in intro multivariable calculus almost always satisfy this.

But if you see a pathological function with discontinuities in the mixed partials, all bets are off. Your professor won't give you those on an exam unless they want to make a point.

The Hessian Matrix

When you arrange all four second order partial derivatives into a matrix, you get the Hessian matrix:

H(f) =

The Hessian shows up everywhere in optimization. The second derivative test for classifying critical points uses the determinant of this matrix. That's its main use in a typical calculus course.

Getting Started: How to Compute Second Order Partials

Let's work through an example. Find all second order partial derivatives of f(x, y) = x³y² + 2xy + sin(x).

Step 1: Find the first order partials

∂f/∂x = 3x²y² + 2y + cos(x)

∂f/∂y = 2x³y + 2x

Step 2: Differentiate each first order partial again

f_xx = ∂/∂x (3x²y² + 2y + cos(x)) = 6xy² - sin(x)

f_yy = ∂/∂y (2x³y + 2x) = 2x³

f_xy = ∂/∂y (3x²y² + 2y + cos(x)) = 6x²y + 2

f_yx = ∂/∂x (2x³y + 2x) = 6x²y + 2

Notice f_xy = f_yx. This function is nice enough that Clairaut's theorem applies.

What These Actually Tell You

f_xx measures curvature in the x-direction. Positive f_xx means the function curves upward as you move in x. Negative means it curves downward.

f_yy does the same for the y-direction.

f_xy and f_yx measure how the slope in one direction changes as you move in the other direction. They're cross-terms. High values mean the function's behavior is strongly coupled between variables.

Common Mistakes

• Forgetting to differentiate all terms in the first partial when computing the second. If a term doesn't contain the variable you're differentiating with respect to, it goes to zero—but only if it truly doesn't contain that variable.
• Mixing up notation. f_xy means differentiate with respect to x first, then y. Not the other way around.
• Assuming mixed partials are equal when they might not be. Check continuity if the problem is ambiguous.
• Calculation errors. Partial derivatives are still derivatives. Chain rule, product rule, quotient rule all still apply.

When You'll Actually Use This

Second order partials show up in:

• Optimization — The second derivative test uses the Hessian's determinant to classify critical points as maxima, minima, or saddle points
• Physics — The wave equation and heat equation involve second order partial derivatives in multiple variables
• Economics — Hessian matrices appear in utility maximization and cost minimization problems
• Machine learning — The Hessian is related to curvature in loss functions during optimization

The math itself isn't complicated. You're just taking derivatives twice. The notation is the main hurdle, and maybe keeping track of which variable you're treating as constant at each step.

Practice with a few functions until the process feels automatic. Then move on to the second derivative test—that's where this stuff actually matters.