Can we split a function while doing differentiation? I have a silly question here. Assume a function $$f(x)= \frac{x}{(x-1)^2} \cdot (x+2)$$
Can I write
$$\frac{d}{dx} f(x) = \frac{d}{dx} x \cdot \frac{d}{dx} (x-1)^{-2} \cdot \frac{d}{dx} (x+2)^{-1}$$ 
If not, then why?

Yes, you can "split" a function; in fact, that's the standard way for computing derivatives algebraically.
However, you have to know what to do with the pieces.
Some examples (the "addition rule", "product rule", and "chain rule") are
$$ \frac{d}{dx} \big( f(x) + g(x) \big) = f'(x) + g'(x) $$
$$ \frac{d}{dx} \big( f(x) \cdot g(x) \big) = f(x) g'(x) + f'(x) g(x) $$
$$ \frac{d}{dx} \big( f(g(x)) \big) = g'(x) \cdot f'(g(x)) $$
