Question

Consider the following demand function with a constant slope. Let Q(p) describe the quantity demanded of the product with respect to price. In this instance Q(p) will take the form Q(p)=a−bp where 0≤p≤ab.
Note: The following is a graph of p(Q) not Q(p).
(diagonal line going down and to the right where p is y and Q is x)
The price elasticity of demand as a function of price is given by the equation E(p)=Q′(p)(p/Q(p)).
Find dE/dp and d^2E/dp^2 (your answers should be in terms of a,b, and p ).

Answer 1

please help

Answer 2

If they mean that
\[ Q_{(p)}' = \frac{d Q_{(p)}}{dp} = -b\]

then
\[E_{(p)}=Q'_{(p)} \left(\frac{p}{Q_{(p)}}\right) = -b\left(\frac{p}{a-bp}\right) = -\frac{bp}{(a-bp)} \]
\[E'_{(p)}= \frac{dE_{(p)}}{dp}=\frac{d}{dp} \frac{-bp}{(a-bp)}\]

Remember the quotient rule
\[ f_{(p)} = \frac{g_{(p)}}{h_{(p)}}\]
\[f'_{(p)} = \frac{g'_{(p)}h_{(p)} - g_{(p)}h'_{(p)}}{[h_{(p)}]^2}\]
\[ g_{(p)} = -bp\]
\[ h_{(p)} = (a-bp)\]

Then
\[\frac{d^2E_{(p)}}{dp^2} = \frac{dE'_{(p)}}{dp} \]