3 - Second-order PDs

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Calculus of Multivariable Functions

Section 3 Second-order Partial Derivatives

The partial derivative of a function of $n$ variables, is itself a function of $n$ variables. By taking the partial derivatives of the partial derivatives, we compute the higher-order derivatives. Higher-order derivatives are important to check the concavity of a function, to confirm whether an extreme point of a function is max or min, etc.

Given that a function $f(x,y)$ is continuously differentiable on an open region, we can derive the following sets of second-order partial derivatives:

Direct second-order partial derivatives:

$f_{xx} = \frac{\partial f_{x}}{\partial x}$ where $f_{x}$ is the first-order partial derivative with respect to $x$.

$f_{yy} = \frac{\partial f_{y}}{\partial y}$ where $f_{y}$ is the first-order partial derivative with respect to $y$.

Cross partial derivatives:

$f_{xy} = \frac{\partial f_{x}}{\partial y}$ where $f_{x}$ is the first-order partial derivative with respect to $x$.

$f_{yx} = \frac{\partial f_{y}}{\partial x}$ where $f_{y}$ is the first-order partial derivative with respect to $y$.

Young's theorem: Corresponding cross partial derivatives are equal. (To read more about Young’s theorem, see Simon & Blume, Mathematics for Economists, p 330.)

Suppose $\color{red}{y = f(x_1,…,x_n)}$ is a continuously differentiable function of $n$ variables.

The first order partial derivative with respect to the variable $\color{red}{x_{i}}$ is $\color{red}{\partial f / \partial x_{i}}$.

The $x_{i}x_{j}$ -second order partial derivative is: $\color{purple}{\frac{\partial }{\partial x_{j}}}(\color{red}{\frac{\partial f}{\partial x_{i}}}) = \frac{\partial^{2}f}{\partial x_{j} \partial x_{i}} = f_{i,j}$

If $\color{red}{j = i}$, then $x_{i}x_{j}$ -second order partial derivative is called $\frac{\partial^{2}f}{\partial x_{i}^2}$ or second order direct partial derivatives.

If $\color{red}{j \neq i}$, then $x_{i}x_{j}$ -second order partial derivative is called the cross partial derivatives.

Example 1: Find the first, second, and cross partial derivatives for the following function: $$f(x,y) = x^{2} + 5xy + 2y^{2}$$ First order partial derivatives: $$f_{x} = 2x + 5y + 0 = 2x + 5y$$ $$f_{y} = 0 + 5x + 4y = 5x + 4y$$ Second order direct partial derivatives: $$f_{xx} = \frac{\partial }{\partial x}(2x + 5y) = 2$$ $$f_{yy} = \frac{\partial }{\partial y}(5x + 4y) = 4$$ Second-order cross partial derivatives: $$f_{xy} = \frac{\partial }{\partial y}(2x + 5y) = 5$$ $$f_{yx} = \frac{\partial }{\partial x}(5x + 4y) = 5$$ See that in this example the cross-partial derivatives are equal.