<s>[INST] If $\frac{x}{y^\frac{n-1}{n}}$ is constant, how do I prove $\frac{dx}{x}=\frac{n-1}{n}\frac{dy}{y}$? From: 
$\frac{x}{y^\frac{n-1}{n}}=constant$
To: 
$\frac{dx}{x}=\frac{n-1}{n}\frac{dy}{y}$
It's from a turbomachinery lecture. The text I'm studying says that binomial expansion is used, but I don't get it.
 [/INST] Let  $x=c\cdot y^\frac{n-1}{n}$. Then 
\begin{align}
\frac{dx}{dy}&=\frac{n-1}{n}c\cdot y^\frac{-1}{n}\\
&=\frac{n-1}{n}c\cdot y^\frac{-1}{n} \cdot \frac{y}{y}\\
&=\frac{n-1}{n}c\cdot y^\frac{n-1}{n} \cdot \frac{1}{y}\\
&=\frac{n-1}{n} \frac{x}{y}\\
\end{align}
</s>