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For questions regarding partial derivatives. The partial derivative of a function of several variables is the derivative of the function with respect to one of those variables, with all others held constant.

The partial derivative of a function of several variables is the derivative of the function with respect to one of the variables, with the others held constant. The partial derivative, like the ordinary derivative, describes the rate of change of a function in a particular direction.

If $f = f(a_1, a_2, \dots, a_n)$ is a function of $n$ variables, then the partial derivative of $f$ with respect to the variable $a_i$ can be written as a limit:

$$\frac{\partial f}{\partial a_i} = \lim_{h \to 0} \frac{f(a_1, \dots, a_i + h, \dots, a_n) - f(a_1, \dots, a_i, \dots, a_n)}{h}.$$

Alternatively, this quantity can be denoted as $f_{a_i}$.

If the function has continuous partial second derivatives, then:

$$\frac{\partial}{\partial x_i}\left(\frac{\partial}{\partial x_j}\right)=\frac{\partial}{\partial x_j}\left(\frac{\partial}{\partial x_i}\right)$$

a result known as Schwarz's Theorem.