Answer
$ \frac{(-0.843x^{-1.1}+0.5x^{-2})(3.2+x^{2.9})-2.9x^{-1.9}(8.43x^{-0.1}-0.5x^{-1}x)}{(3.2+x^{2.9})^{2}}$
Work Step by Step
$\displaystyle \frac{d}{dx}[\frac{f(x)}{g(x)}]= \displaystyle \frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{[g(x)]^{2}}$
$f(x)=8.43x^{-0.1}-0.5x^{-1},$
$f^{\prime}(x)=8.43(-0.1x^{-1.1})-0.5(-x^{-2})$
$=-0.843x^{-1.1}+0.5x^{-2}$
$g(x)=3.2+x^{2.9}$
$g^{\prime}(x)=2.9x^{-1.9}$
$ \frac{dy}{dx}=\frac{(-0.843x^{-1.1}+0.5x^{-2})(3.2+x^{2.9})-(8.43x^{-0.1}-0.5x^{-1}x)(2.9x^{-1.9})}{(3.2+x^{2.9})^{2}}$
... we do not need to expand the answer...