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The derivative of the demand function \( x \) with respect to the price \( p \) is:

\[
\frac{dx}{dp} = \frac{4p}{(p + 2)^2} - 1 - \frac{4}{p + 2}
\]

Now, let's compute the rate of change at \( p = 5 \).It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can substitute \( p = 5 \) directly into the derivative equation:

\[
\frac{dx}{dp} = \frac{4(5)}{(5 + 2)^2} - 1 - \frac{4}{5 + 2}
\]

This will give you the rate of change in the demand \( x \) when the price \( p \) is 5.

Would you like further clarification on this, or have any other questions? Here are some related questions:

1. How does the demand function change as the price increases or decreases?
2. What is the economic interpretation of \(\frac{dx}{dp}\)?
3. How does the presence of the term \(\frac{4p}{p + 2}\) affect the demand?
4. What does it mean for the rate of change to be negative?
5. How would the rate of change vary if the price \( p \) were different?

**Tip:** When dealing with derivatives in economics, the sign of the derivative often tells you whether a quantity is increasing or decreasing with respect to the variable.

Consider the following demand function with demand x and price p. x = 300 - p - (4p / (p + 2)). Find dx/dp and the rate of change in the demand x for the given price p = 5.

Learn how to calculate the rate of change in a demand function x = 300 - p - (4p / (p + 2)) with respect to price p. This problem includes differentiation and substitution of p = 5 into the derivative formula. Suitable for advanced high school or college calculus students.

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