The ideal gas law states that for a given quantity of confined gas in a container, the pressure P, volume V and temp T satisfy an equation $P=kT/V$, where k is a positive constant. Show that:
$$V \frac{\partial P}{\partial V} = -P$$
and
$$V \frac{\partial P}{\partial V}+T\frac{\partial P}{\partial T} = 0.$$
I would appreciate anyones explanation to the two parts of this question, after many tries the closest I can come to is =P for the first part and cannot find my mistake.
More generally, let $y=cx^\alpha$. Then
$$ x\frac{\partial y}{\partial x}=xc\alpha x^{\alpha-1}=\alpha cx^\alpha=\alpha y\;. $$
In your case, we have $P\sim V^{-1}$ and $P\sim T^1$, so
$$ V\frac{\partial P}{\partial V}+T\frac{\partial P}{\partial T}=-1\cdot P+1\cdot P=0\;. $$