I have a little problem trying to understand differentials

Question

I have just learned differentials and I have a few things unclear. For example I found a contradiction trying to differentiate the function $f(f(x))$, where $f:\mathbb{R} \to \mathbb{R}$. Using the chain rule we get $$(f(f(x)))'=f'(f(x))f'(x)$$ However, if we use the differentials we get that $$\frac{d(f(f(x)))}{dx}=\frac{df}{df}\frac{df}{dx}=\frac{df}{dx}$$ This is true if $f(x)$ is a constant or $f(f(x))=f(x)$. Why doesn't this work for the general case?