The demand function for a certain brand of compact discs is given by

p = −5x^2 − 5x + 36

where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.


(b) Use implicit differentiation to compute the rate of change of demand with respect to price, p, when x = 2. Do not round your answer.
Rate of change of demand, x' =
thousands of units per dollar


(c) Compute the elasticity of demand when x = 2. Do not round your answer.
Elasticity of Demand =

Let's work through the problem step by step.

### Part (b): Rate of change of demand with respect to price using implicit differentiation

We are given the demand function:

\[
p = -5x^2 - 5x + 36
\]

where \( p \) is the price and \( x \) is the quantity demanded (in thousands of units). We are asked to compute the rate of change of demand with respect to price, which is \( \frac{dx}{dp} \), when \( x = 2 \).

#### Implicit Differentiation:
To find \( \frac{dx}{dp} \), we differentiate the given demand function implicitly with respect to \( p \).

First, take the derivative of the equation with respect to \( x \) on both sides:

\[
\frac{dp}{dx} = \frac{d}{dx}(-5x^2 - 5x + 36)
\]

The derivative of the right-hand side with respect to \( x \) is:

\[
\frac{dp}{dx} = -10x - 5
\]

Now, the rate of change of demand with respect to price is the reciprocal of \( \frac{dp}{dx} \):

\[
\frac{dx}{dp} = \frac{1}{\frac{dp}{dx}} = \frac{1}{-10x - 5}
\]

#### Evaluate at \( x = 2 \):
Now, substitute \( x = 2 \) into the derivative expression:

\[
\frac{dx}{dp} = \frac{1}{-10(2) - 5} = \frac{1}{-20 - 5} = \frac{1}{-25}
\]

Thus, the rate of change of demand with respect to price when \( x = 2 \) is:

\[
\frac{dx}{dp} = -\frac{1}{25} \text{ thousands of units per dollar}
\]

### Part (c): Elasticity of demand

The elasticity of demand \( E(x) \) is given by the formula:

\[
E(x) = \frac{dx}{dp} \times \frac{p}{x}
\]

We already know that \( \frac{dx}{dp} = -\frac{1}{25} \) when \( x = 2 \).

Next, we need to find \( p \) when \( x = 2 \). Substitute \( x = 2 \) into the demand function:

\[
p = -5(2)^2 - 5(2) + 36 = -5(4) - 10 + 36 = -20 - 10 + 36 = 6
\]

So, when \( x = 2 \), \( p = 6 \).

Now, compute the elasticity:

\[
E(x) = \left( -\frac{1}{25} \right) \times \frac{6}{2} = -\frac{1}{25} \times 3 = -\frac{3}{25}
\]

Thus, the elasticity of demand when \( x = 2 \) is:

\[
E(x) = -\frac{3}{25}
\]

### Summary of Answers:
(b) The rate of change of demand with respect to price when \( x = 2 \) is \( -\frac{1}{25} \) thousands of units per dollar.

(c) The elasticity of demand when \( x = 2 \) is \( -\frac{3}{25} \).

---

Would you like more details or further explanation on any of the steps? 

Here are 5 related questions you might find useful:
1. How does the elasticity of demand relate to price sensitivity?
2. What is the meaning of a negative elasticity value?
3. How can we graph the demand curve from this equation?
4. How does the rate of change of demand affect pricing strategies?
5. What does it mean for a demand curve to be elastic versus inelastic?

**Tip:** Elasticity of demand helps determine how a change in price will affect the total revenue from sales. If the elasticity is greater than 1 in magnitude, demand is elastic, meaning a price decrease can increase total revenue.

The demand function for a certain brand of compact discs is given by p = −5x^2 − 5x + 36 where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. (b) Use implicit differentiation to compute the rate of change of demand with respect to price, p, when x = 2. Do not round your answer. Rate of change of demand, x' = thousands of units per dollar. (c) Compute the elasticity of demand when x = 2. Do not round your answer. Elasticity of Demand =

Learn how to apply implicit differentiation to compute the rate of change of demand with respect to price and the elasticity of demand for a specific product. This problem involves a quadratic demand function and is suitable for students in Grades 11-12 studying calculus and economics.

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