The demand function for drangles is $q(p)=(p+1{)}^{-2}.$

$($a$)$ What is the price elasticity of demand at price $p?.$

$($b$)$ At what price is the price elasticity of demand for drangles equal to $-1?$

$($c$)$ Write an expression for total revenue from the sale of drangles as a function of their price. Use calculius to find the revente$-$maximizing price, Don't forget to check the second$-$order condition.

$($d$)$ Suppose that the demand function for drangles takes the more general form $q(p)=(p+a{)}^{-b}$ where $a>0$ and $b>1.$ Calculate an expression for the price elasticity of demand at price $p.$ At what price is the price elasticity of demad equal to $-1?$