Variables change in derivation. What is wrong? I have two set of variables, which are related: $\left\{ \alpha, \beta, \gamma \right\}$ and $\left\{ v_0, v_1, v_2 \right\} = \left\{ \alpha, \beta, \beta \gamma \right\}$ 
Now, I want to compute the partial derivative $\frac{\partial \beta}{\partial \left(\alpha \beta \gamma \right)}$. 
My first approach: as $\alpha$ and $\gamma$ do not depend on $\beta$, 
$\frac{\partial \beta}{\partial \left(\alpha \beta \gamma \right)} = \frac{1}{\alpha  \gamma} \frac{\partial \beta}{\partial \beta} = \frac{1}{\alpha  \gamma} = \frac{v_1}{v_0 v_2}$. 
But also, one could thinks as follows:
$\frac{\partial \beta}{\partial \left(\alpha \beta \gamma \right)} =  \frac{\partial v_1}{\partial \left( v_0 v_2 \right)} = 0$
Something should be meshed up related with the dependencies, but I cannot see it. Can anyone give me a hand?
 A: In your second approach, you write
$$\frac{\partial \beta}{\partial \left(\alpha \beta \gamma \right)} = \frac{\partial v_1}{\partial \left( v_0 v_2 \right)} = 0 \tag{1}\label{eq1}$$
This only works if $v_1$ doesn't change as $v_0 v_2$ does, which is what I believe you are assuming. However, since $v_0 v_2 = \alpha \beta \gamma$ and $v_1 = \beta$, the $2$ expressions are not generally independent of each other due to $\beta$ being in both expressions (note the exception is if $\alpha \gamma$ has a dependency on $\beta$ only as a fraction with a factor of $\beta$ in the denominator, but you say "as $\alpha$ and $\gamma$ do not depend on $\beta$", then this is not true). Thus, if for example, you change $v_0 v_2$ by changing $\beta$, then $v_1$ will be changing as well, so you cannot state that $v_1$ doesn't change at all while $v_0 v_2$ is changing, so this means that
$$\frac{\partial v_1}{\partial \left( v_0 v_2 \right)} \neq 0 \tag{2}\label{eq2}$$
More generally, note that expressing a value in a different way doesn't change any result, or other behavior, which uses that value. For example, $6 = 3 \times 2$, but $\frac{30}{6}$ and $\frac{30}{3 \times 2}$ are both equal to $5$. The important thing is what particular numeric values are being used, regardless of how they're represented. In this situation, expressing $\beta$ as $v_1$ and $\alpha \beta \gamma$ as $v_2$ doesn't change their numeric values, or relationship, at any point, so their extent of (in)dependence is still the same, and any partial derivatives involving them are also the same.
I hope that I've interpreted what you are doing correctly and that this addresses the issue.
A: I do not see the role of the second set of variables. If I intepret correctly, you are trying to compute $\frac{\partial \beta}{\partial(\alpha,\beta,\gamma)}$, that is the vector $(0,1,0)$.
