<s>[INST] How to differentiate $y=(x+1)^3/x^{3/2}$ and $y=2x^4/(b^2-x^2)$ I need to solve a list of derivatives to help me on an exam; however, I'm in doubt when they use another variable (constant) or when I have a fraction with functions that use the power rule.
For example:
$$y = \frac{(x + 1)^3}{x^{3/2}}.$$
This one I tried having $u = x + 1$, deriving it and having $v = x^{3/2}$, but I couldn't get the right answer.
And:
$$y = \frac{2x^4}{b^2 - x^2}.$$
And this one I tried to separate the constant, like $y = \frac{1}{b^2} \frac{dx}{dy} \frac{2x^4}{-x^2}$, but also coudn't find the answer.
Can you give a tip on how to find which rule to apply to these kind of derivatives?
 [/INST] You really only need the quotient rule and the power rule.
For the quotient rule, when $y=\frac{f(x)}{g(x)}$ (where $g(x)\neq 0$), we have $y'=\frac{f'g-fg'}{g^2}$.
Problem 1: Differentiate $y = \frac{(x + 1)^3}{x^{3/2}}$ with respect to $x$.
Solution. We have $y=\frac{f(x)}{g(x)}$ where $f(x)=(x+1)^3$ and $g(x)=x^{3/2}$. Thus, we also have $f'(x)=3(x+1)^2$ and $g'(x)=\frac{3}{2}x^{1/2}$. Using the quotient rule, we see that
$$
y' = \frac{3(x+1)^2\cdot x^{3/2}-(x+1)^3\cdot\frac{3}{2}x^{1/2}}{(x^{3/2})^2}.
$$
Problem 2: Differentiate $y = \frac{2x^4}{b^2 - x^2}$ with respect to $x$.
Solution. We have $y=\frac{f(x)}{g(x)}$ where $f(x)=2x^4$ and $g(x)=b^2-x^2$. Thus, we also have $f'(x)=8x^3$ and $g'(x)=-2x$. Using the quotient rule, we see that
$$
y'=\frac{8x^3\cdot(b^2-x^2)-2x^4\cdot(-2x)}{(b^2-x^2)^2}.
$$
If required, all that is left to do is for you to simplify. 
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