Given the function q=D(x)=k/x^n
where k is a positive constant and n is an integer greater than 0.
How would you find the elasticity of the demand function
After you find the der. what do you do?
To find the elasticity of the demand function, we need to differentiate the function q = D(x) with respect to x. Let's go through the steps:
1. Start with the demand function q = D(x) = k/x^n.
2. Take the derivative of the demand function with respect to x, denoted as D'(x):
D'(x) = (-kn)/x^(n+1).
3. Now, the elasticity of demand is defined as the percentage change in quantity demanded (q) divided by the percentage change in price (p). In this case, we are finding the elasticity with respect to price (p). Therefore, we need to express the function in terms of price (q) rather than quantity (x).
4. The price (p) is the reciprocal of the quantity (x), so we can rewrite the demand function as: q = k/p^n.
5. Differentiate the demand function with respect to p, denoted as D'(p):
D'(p) = (-kn)/p^(n+1).
6. Now, we have the derivative D'(x) with respect to x and D'(p) with respect to p.
7. To find the elasticity of demand, we use the formula:
Elasticity = (D'(p) * p) / q.
8. Plug in the values of D'(p), p, and q derived from the previous steps into the formula to obtain the elasticity of the demand function.
After finding the derivative, the next step is to use the derived derivative to solve any specific problem or question related to the demand function. For example, you can use the elasticity of demand to analyze the responsiveness of demand to changes in price or to compare the elasticities of different demand functions to determine their relative sensitivity to price changes.