I am trying to determine how to use differentials to estimate the percentage change in $r$, if $x$ increases by 6%. Let $r=6x^{-1/6}, x>0$.
So far, I have done the following steps:
1) Determine the relative change in $x$, which is $0.06$.
2) Find the differential of the equation. So, $dr = -x^{-7/6}*(dx)$
3) Divide both sides by $r$, which gives $(x^{1/6}(dx))/(-6x^{7/6})$ or $(dx)/(-6x)$
4) $(dx)/x = 0.06$, so, multiply $(-1/6)$ with $0.06$ to get $-0.01$.
5) The percentage change in r is $-1%$%
A few questions, however:
In step three, why do we divide both sides by r?
In step five, why is $(dx)/x = 0.06$?