Higher-order partial derivatives describe how the rate of change of a multivariable function changes as you move in one coordinate direction while holding the others fixed. For a surface z = f(x,y), first partial derivatives measure slope in the x and y directions, while second partial derivatives measure bending or curvature. These ideas matter in physics, engineering, economics, and machine learning because many real systems depend on more than one input.
They help identify peaks, valleys, saddle points, and how small changes propagate through a model.
The pure second partials f_xx and f_yy measure how the x-slope changes with x and how the y-slope changes with y. The mixed partials f_xy and f_yx measure how the slope in one direction changes as you move in the other direction. When f_xy and f_yx are continuous near a point, Clairaut's theorem says they are equal, which is a powerful simplification.
A common way to organize all second partial derivatives is the Hessian matrix, which is used to classify critical points and approximate surfaces locally.
Key Facts
- First partial derivatives: f_x = ∂f/∂x and f_y = ∂f/∂y.
- Pure second partial derivatives: f_xx = ∂²f/∂x² and f_yy = ∂²f/∂y².
- Mixed partial derivatives: f_xy = ∂/∂y(f_x) and f_yx = ∂/∂x(f_y).
- Clairaut's theorem: if f_xy and f_yx are continuous near a point, then f_xy = f_yx at that point.
- Hessian matrix for f(x,y): H = [[f_xx, f_xy], [f_yx, f_yy]].
- Second derivative test: D = f_xx f_yy - (f_xy)^2 helps classify critical points when f_x = 0 and f_y = 0.
Vocabulary
- Partial derivative
- A derivative of a multivariable function with respect to one variable while the other variables are held constant.
- Higher-order partial derivative
- A partial derivative found by differentiating a function more than once with respect to one or more variables.
- Mixed partial derivative
- A second or higher derivative taken with respect to different variables, such as f_xy or f_yx.
- Clairaut's theorem
- A theorem stating that mixed partial derivatives are equal when they are continuous near the point being studied.
- Hessian matrix
- A square matrix containing all second partial derivatives of a multivariable function.
Common Mistakes to Avoid
- Treating other variables as constants incorrectly. When finding f_x, y must be treated like a constant number, not like a function of x.
- Confusing f_xy with f_xx. The notation f_xy means differentiate first with respect to x and then with respect to y, not differentiate twice with respect to x.
- Assuming mixed partials are always equal. Clairaut's theorem requires continuity of the mixed partial derivatives near the point, so equality is not guaranteed for every function.
- Using the second derivative test before finding a critical point. The test with D = f_xx f_yy - (f_xy)^2 applies only at points where f_x = 0 and f_y = 0.
Practice Questions
- 1 For f(x,y) = x^3y^2 + 4xy - y^3, find f_x, f_y, f_xx, f_yy, f_xy, and f_yx.
- 2 For f(x,y) = e^(2x)y^3, compute f_xy and f_yx at the point (0,2).
- 3 A function has continuous second partial derivatives near a point and f_xy = 7 there. Explain what Clairaut's theorem tells you about f_yx at the same point and why continuity matters.