the diversity of a population that contains two species can be measured by the gini - simpson diversity…
the diversity of a population that contains two species can be measured by the gini - simpson diversity index. if the fraction of organisms from species 1 is p, th the diversity is given by h(p) below. use the product rule to calculate h(p). for wha value of p does h(p) = 0?\nh(p)=2p(1 - p)\nh(p)=0
Answer
Explanation:
Step1: Apply the product rule
The product rule states that if (y = uv), then (y^\prime=u^\prime v + uv^\prime). Let (u = 2p) and (v=(1 - p)). Then (u^\prime=2) and (v^\prime=- 1). [ \begin{align*} H^\prime(p)&=(2p)^\prime(1 - p)+2p(1 - p)^\prime\ \end{align*} ]
Step2: Calculate the derivative
Substitute (u^\prime = 2) and (v^\prime=-1) into the product - rule formula. [ \begin{align*} H^\prime(p)&=2(1 - p)+2p(-1)\ &=2-2p - 2p\ &=2-4p \end{align*} ]
Step3: Solve (H^\prime(p)=0)
Set (H^\prime(p)=2 - 4p=0). [ \begin{align*} 2-4p&=0\ -4p&=-2\ p&=\frac{-2}{-4}=\frac{1}{2} \end{align*} ]
Answer:
(H^\prime(p)=2 - 4p) and (p = \frac{1}{2}) when (H^\prime(p)=0)