I have just seen this notation of a question: Find $$\frac{d(x-x\sin(x))}{d(1-\cos(x))}$$ or something along those lines.
I am well aware of notation like $\frac{dy}{dx}$ or something like $\frac{d(\sin(x))}{dx}$ but I don't really know what the above means. I can't remember if that was the question exactly but I doubt it's of much importance.
Any help on how to evaluate derivatives like this and what it means.
It simply says: Differentiate $x-x\sin(x)$ with respect to $1-\cos(x)$. It does seem rather weird to differentiate such an expression, but the way to progress is through the chain rule. That is, $$\frac{d(x(1-\sin(x))}{d(1-\cos(x))}=\frac{d(x(1-\sin(x))}{dx}\times \frac{dx}{d(1-\cos(x))}.$$
Therefore the expression simplifies to
$$=[x\times (-\cos(x))+1-\sin(x)]\times \frac{1}{\sin(x)}.$$ $$=\frac{-(x\cos(x)+\sin(x))+1}{\sin(x)}$$
