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Chain Rule & Demand Function

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Please show all the steps to calculate the answers to the attached questions about using the chain rule on an American population function and a demand function for desk lamp prices.

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https://brainmass.com/math/functional-analysis/chain-rule-demand-function-574527

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The solution gives detailed steps on determining derivative and using the derivative (chain rule) to find the rate of change. All formula and calculations are shown and explained.

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Properties of the equilibrium price for a demand function

The demand function for a certain good is D(p,m) and the supply function is S(p). For a given m, the equilibrium price p* is given by p* = f(m).

a) Show that

f'(m) = [dD(p*,m)/dm] / [S'(p*) - dD(p*,m)/dp*]

b) Verify this result when S(p) = 2p and D(p,m) = 6m^2p^(-1) + m

c) By using the implicit function rule, or otherwise, show how the result in (a) may be modified if the supply function depends on both p and m.

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