Answer
$h^{\prime}(x)=-\displaystyle \frac{6x}{(x^{2}+1)^{4}}$
Work Step by Step
We can rewrite $h(x)=(x^{2}+1)^{-3}$
Let $u(x)=x^{-3},\qquad v(x)=x^{2}+1$
$\displaystyle \frac{du}{dx}=-3x^{-4}, \quad \frac{dv}{dx}=2x$
Then,$\quad y=h(x)=u(v(x))$ and $f^{\prime}(x)=\displaystyle \frac{dy}{dx}=\frac{du}{dv}\frac{dv}{dx}$
$\displaystyle \frac{du}{dv}=-3v^{-4}=-3(x^{2}+1)^{-4}$
$h^{\prime}(x)=\displaystyle \frac{du}{dv}\frac{dv}{dx}=-3(x^{2}+1)^{-4}\cdot(2x)$
$h^{\prime}(x)=-6x(x^{2}+1)^{-4}$
$h^{\prime}(x)=-\displaystyle \frac{6x}{(x^{2}+1)^{4}}$
