Answer
$\displaystyle \mathrm{r}^{\prime}(t)=-\frac{1}{(1+t)^{2}}\mathrm{i}+\frac{1}{(1+t)^{2}}\mathrm{j}+\frac{t^{2}+2t}{(1+t)^{2}}\mathrm{k}$
Work Step by Step
Parametric equations: $\mathrm{r}(t):\quad\left\{\begin{array}{ll}
x=(1+t)^{-1} & \\
y=t(1+t)^{-1} & \\
y=t^{2}(1+t)^{-1} &
\end{array}\right.$
Differentiate $(\displaystyle \frac{d}{dt})$ each component function.
x: chain rule
y,z: quotient rule
$\mathrm{r}^{\prime}(t):\quad\left\{\begin{array}{ll}
x=-1(1+t)^{-2}(1) & =-\dfrac{1}{(1+t)^{2}}\\ \\
y=\dfrac{(1+t)\cdot 1-t(1)}{(1+t)^{2}} & =\dfrac{1}{(1+t)^{2}}\\\\
z=\dfrac{(1+t)\cdot 2t-t^{2}(1)}{(1+t)^{2}} & =\dfrac{t^{2}+2t}{(1+t)^{2}}
\end{array}\right.\\ $
$\displaystyle \mathrm{r}^{\prime}(t)=-\frac{1}{(1+t)^{2}}\mathrm{i}+\frac{1}{(1+t)^{2}}\mathrm{j}+\frac{t^{2}+2t}{(1+t)^{2}}\mathrm{k}$