Solution to $\frac{d}{d\frac{1}{x}} x$
$\frac{\mathrm{d} x}{\mathrm{d} \frac{1}{x}}=\left (\frac{\mathrm{d} \frac{1}{x}}{\mathrm{d} x} \right )^{-1}=\left ( \frac{-1}{x^{2}} \right )^{-1}=-x^{2}$
$x=(\frac{1}{x})^{-1}$. Set $y=\frac{1}{x}$ and use ordinary Differentiation rules. Then express $y$ in Terms of $x$.