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?

You can make life easier using logarithmic differentiation since you just have here product, quotient and powers$$f= \frac{x (x+2)}{(x-1)^2}\implies \log(f)=\log(x)+\log(x+2)-2\log(x-1)   $$ Differentiate both sides
$$\frac {f'}f=\frac 1x+\frac 1{x+2}-\frac 2{x-1}$$ Simplify and, when done, use
$$f'=f \times \left(\frac {f'}f \right)$$ and simplify again.
