Solution to $\frac{d}{d\frac{1}{x}} x$ If I want to solve 
$$\frac{d}{d\frac{1}{x}} x$$
is my approach correct?
As
$$\begin{align*} \frac{d}{d\frac{1}{x}}x&=\\
\text{with }\frac{1}{x}&=y\\
\frac{d}{dy}\frac{1}{y}&=-\frac{1}{y^2}\\
&=-\frac{1}{\left(\frac{1}{x}\right)^2}\\
&=-x^2
\end{align*}$$
Is this approach correct, or did I miss something?

$\frac{\mathrm{d} x}{\mathrm{d} \frac{1}{x}}=\left (\frac{\mathrm{d} \frac{1}{x}}{\mathrm{d} x}  \right )^{-1}=\left ( \frac{-1}{x^{2}} \right )^{-1}=-x^{2}$
