以下是一个相对复杂的数学公式示例，涉及多元函数的偏导数与复合函数求导（以三元函数为例）： 设有三元函数$z = f(x, y, u)$，其中$u = g(x, v)$，$v = h(x, y)$。 要求$\frac{\partial z}{\partial x}$和$\frac{\partial z}{\partial y}$。 根据复合函数求导法则： 对于$\frac{\partial z}{\partial x}$： 首先，将$y$看作常数，对$z = f(x, y, u)$关于$x$求偏导，得到$\frac{\partial z}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}$。 又因为$u = g(x, v)$，这里$v = h(x, y)$，再次使用复合函数求导法则，有$\frac{\partial u}{\partial x}=\frac{\partial g}{\partial x}+\frac{\partial g}{\partial v}\frac{\partial v}{\partial x}$。 而$\frac{\partial v}{\partial x}=\frac{\partial h}{\partial x}$。 所以$\frac{\partial u}{\partial x}=\frac{\partial g}{\partial x}+\frac{\partial g}{\partial v}\frac{\partial h}{\partial x}$。 将其代入$\frac{\partial z}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}$中，可得： $\frac{\partial z}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial u}(\frac{\partial g}{\partial x}+\frac{\partial g}{\partial v}\frac{\partial h}{\partial x})$。 同理，对于$\frac{\partial z}{\partial y}$： 先固定$x$，对$z = f(x, y, u)$关于$y$求偏导，得到$\frac{\partial z}{\partial y}=\frac{\partial f}{\partial y}+\frac{\partial f}{\partial u}\frac{\partial u}{\partial y}$。 由于$u = g(x, v)$，$v = h(x, y)$，所以$\frac{\partial u}{\partial y}=\frac{\partial g}{\partial y}+\frac{\partial g}{\partial v}\frac{\partial v}{\partial y}$。 而$\frac{\partial v}{\partial y}=\frac{\partial h}{\partial y}$。 则$\frac{\partial u}{\partial y}=\frac{\partial g}{\partial y}+\frac{\partial g}{\partial v}\frac{\partial h}{\partial y}$。 代入可得$\frac{\partial z}{\partial y}=\frac{\partial f}{\partial y}+\frac{\partial f}{\partial u}(\frac{\partial g}{\partial y}+\frac{\partial g}{\partial v}\frac{\partial h}{\partial y})$。 综上，通过层层嵌套的复合函数求导过程，得到了多元函数关于不同自变量的偏导数表达式，展现了较为复杂的数学公式推导过程。