Answer
$h'(r)=\dfrac{r-6\sqrt{r}-1}{2\sqrt{r}(r+1)^{2}}$
Work Step by Step
$h(r)=\dfrac{2-r-\sqrt{r}}{r+1}$
Rewrite the square root using a rational exponent:
$h(r)=\dfrac{2-r-\sqrt{r}}{r+1}=\dfrac{2-r-r^{1/2}}{r+1}$
Evaluate the derivative using the quotient rule:
$h'(r)=\dfrac{(r+1)(2-r-r^{1/2})'-(2-r-r^{1/2})(r+1)'}{(r+1)^{2}}=...$
Evaluate the derivatives indicated in the numerator:
$...=\dfrac{(r+1)\Big(-1-\dfrac{1}{2}r^{-1/2}\Big)-(2-r-r^{1/2})(1)}{(r+1)^{2}}=...$
Simplify:
$...=\dfrac{(r+1)\Big(-1-\dfrac{1}{2r^{1/2}}\Big)-2+r+r^{1/2}}{(r+1)^{2}}=...$
$...=\dfrac{-r-\dfrac{r}{2r^{1/2}}-1-\dfrac{1}{2r^{1/2}}-2+r+r^{1/2}}{(r+1)^{2}}=...$
$...=\dfrac{-\Big(\dfrac{r+1}{2r^{1/2}}\Big)-3+r^{1/2}}{(r+1)^{2}}=\dfrac{\dfrac{-(r+1)-6r^{1/2}+2r}{2r^{1/2}}}{(r+1)^{2}}=...$
$...=\dfrac{-r-1-6r^{1/2}+2r}{2r^{1/2}(r+1)^{2}}=\dfrac{r-6r^{1/2}-1}{2r^{1/2}(r+1)^{2}}=...$
Change the rational exponents to square roots:
$...=\dfrac{r-6\sqrt{r}-1}{2\sqrt{r}(r+1)^{2}}$
