While working through differential calculus questions for the chain rule, I stumbled upon:
$$y = \left(\frac{x}{1-\sqrt{x}}\right)^3 $$
I initially attempted to apply the chain rule, but to apply it, I would need to differentiate the contents in the brackets, which, from what I know, I can only differentiate using the quotient rule. However, in my book, the quotient rule is taught later and thus I would assume that I can only use the mathematical tools taught thus far, i.e. the chain rule and the 'differentiating short-cut' (that's what my teacher calls it), i.e. if $f(x) = ax^n$, $f'(x) = anx^{n-1}$. I cannot figure out a way to solve this question by only using only the chain rule and differentiating short-cut; am I missing something or is the question simply in the wrong place in my book?
I would also like to point out that when I looked at the worked solutions for this maths book, they had all answers for this exercise except for that question. The solutions displayed without working is:
$$\frac{1-2\sqrt{x}}{4\sqrt{x-x\sqrt{x}}}$$
Even if I were to use the quotient rule and chain rule, I get a different answer (I even repeated my working twice in case I made a mistake, but I got the same answer both times):
$$\frac{3x^2-\frac{3}{2}x^{\frac{5}{2}}}{(1-\sqrt{x})^4}$$
EDIT: I believe the product rule can also not be used to solve this question as it is, just like the quotient rule, taught later in the book.
Bibliography:
Mathematics Higher Level, IB, by Josip Harcet et. al.