I'm blanking on the simplest thing ever:
$$L = x$$ $$\frac{dL}{dx} = 1$$
But if I do:
$$x = m + n$$ $$\frac{dL}{dx} = \frac{\partial L}{\partial m}\frac{\partial m}{\partial x} + \frac{\partial L}{\partial n}\frac{\partial n}{\partial x}$$
$$\frac{dL}{dx} = (1)(1) + (1)(1)$$ $$\frac{dL}{dx} = 2$$
Yeah, this is embarrassing.