I am trying to calculate $\frac{\partial (xy^2)}{\partial (xy)}$.
To calculate this I am trying to substitute expressions in $\alpha, \beta$ instead of $x, y$ in two independent methods.
Method-1:
Suppose $x = \sqrt{\alpha \beta}$ and $y = \sqrt{\frac{\alpha}{\beta}}$.
So $xy^2 = {\alpha}^{3/2}{\beta}^{-1/2}$ and $xy = \alpha$.
$\frac{\partial (xy^2)}{\partial (xy)}$ = $\frac{\partial ({\alpha}^{3/2}{\beta}^{-1/2})}{\partial \alpha}$ $= {\beta}^{-1/2} \cdot \frac{3\alpha^{1/2}}{2} = \frac{3y}{2}$
Method-2:
Suppose $x = \frac{\alpha}{\sqrt{\beta}}$ and $y = \sqrt{\beta}$
$\frac{\partial (xy^2)}{\partial (xy)}$ = $\frac{\partial (\alpha \sqrt{\beta})}{\partial (\alpha)} = \sqrt{\beta} = y $
What is wrong about the methods I am using? After all it is basic substitution.
EDIT: I see. The actual motivation for the question was a problem in 2-D vector calculus. I wanted to calculate $div(\overrightarrow{v})$ in polar coordinates. So is it wrong to write:
$\frac{\partial{v_x}}{\partial{x}} $ = $ \frac{\partial ({v_r cos\theta - v_{\theta} sin\theta})} {\partial({r cos\theta})}$
where $ v_r$ and $v_{\theta}$ are components of $\overrightarrow{v}$ along $\overrightarrow{r}$ and $\overrightarrow{\theta}$