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.