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?

When the expression contains products and ratioé, logarithmic differentiation make life much easier.
Let us consider the first problem $$y = \frac{(x + 1)^3}{x^{3/2}}$$ Taking logarithms $$\log(y)=3\log(x+1)-\frac32\log(x)$$ Now, differentiate both sides $$\frac{y'}y=\frac3{x+1}-\frac3{2x}=\frac{3 (x-1)}{2 x (x+1)}$$ $$y'=\frac{3 (x-1)}{2 x (x+1)}y=\frac{3 (x-1)}{2 x (x+1)} \frac{(x + 1)^3}{x^{3/2}}=\frac{3 (x-1) (x+1)^2}{2 x^{5/2}}$$
Do the same for the second problem $$y = \frac{2x^4}{b^2 - x^2}$$ $$\log(y)=\log(2)+4\log(x)-\log(b^2-x^2)$$ $$\frac{y'}y=0+\frac 4x+\frac{2x}{b^2-x^2}=\frac{2(2b^2-x^2)}{x(b^2-x^2)}$$ $$y'=\frac{2(2b^2-x^2)}{x(b^2-x^2)}y=\frac{2(2b^2-x^2)}{x(b^2-x^2)}\frac{2x^4}{b^2 - x^2}=\frac{4(2b^2-x^2)x^3}{(b^2-x^2)^2}$$
