Question

In Exercises, find \(d x / d p\) for the demand function. Interpret this rate of change when the price is \(\$ 10\). $$ x=\ln \frac{1000}{p} $$

Short answer

The rate of change of the quantity demanded with respect to the price is \(-\frac{1}{p}\). When the price is $10, the quantity demanded decreases by 0.1 units for every $1 increase in price.

Step-by-step solution

Differentiate the function with respect to \(p\)

Start by differentiating \(x = \ln \frac{1000}{p}\), one can use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outside function multiplied by the derivative of the inside function. Here, the outside function is the natural logarithm and inside function is \(\frac{1000}{p}\). \n\n The derivative of \(x\) with respect to \(p\) is given by the quotient \(-\frac{1000}{p}\) times \(1 / \frac{1000}{p}\), simplifying this gives : \(-\frac{1}{p}\).

Interpret the rate of change at \(p = $10\)

Substitute \(p = 10\) into the derivative \(-\frac{1}{p}\). This results in \(-\frac{1}{10}\). In economic terms, this value is understood as for every $1 increase in price, the quantity demanded decreases by 0.1 units.

Key concepts

Demand Function

The demand function is a fundamental concept in economics. It describes the relationship between the price of a good and the quantity demanded by consumers. When the price of a product changes, the demand function helps predict how the demand will adjust. Understanding this function is essential for businesses to set prices that maximize profits.

• The demand function can be expressed using various mathematical expressions, making it versatile in different scenarios.
• In this exercise, the specific demand function is given by the expression: \( x = \ln \frac{1000}{p} \), where \( x \) is the quantity demanded and \( p \) is the price.
• Knowing how to manipulate and differentiate these expressions can provide valuable insights into how sensitive demand is to price changes.

Chain Rule

The Chain Rule is a powerful tool in calculus used to differentiate composite functions. A composite function is a function that is made up of two or more simpler functions, each nested within the other. By understanding how to apply the Chain Rule, you can find derivatives of these more complex functions.

• The Chain Rule states: If a function \( y = f(g(x)) \) is a combination of two functions \( f \) and \( g \), then its derivative is \( f'(g(x)) \cdot g'(x) \).
• In this demand function, the composite nature arises because \( x = \ln(\frac{1000}{p}) \), where the natural logarithm \( \ln \) is the outer function and \( \frac{1000}{p} \) is the inner one.
• Applying the Chain Rule here helps find \( \frac{dx}{dp} \), which is the rate at which the quantity demanded changes as the price changes.

Rate of Change

The rate of change is a key concept in calculus that measures how a quantity changes relative to another. In the context of a demand function, it tells us how the quantity demanded changes as the price changes.

• For the given demand function, we found that the derivative, \( \frac{dx}{dp} = -\frac{1}{p} \), represents the rate of change of demand concerning price.
• This derivative shows that for a small increase in price, the quantity demanded decreases. More specifically at \( p = 10 \), the rate of change \( \frac{dx}{dp} = -0.1 \) signifies that a $1 increase in price leads to a decrease of 0.1 unit in demand.
• Understanding this rate helps businesses make informed pricing decisions by anticipating demand fluctuations.

Natural Logarithm

The natural logarithm is a crucial mathematical function frequently used in calculus and various real-life applications like growth models and economics. It is represented as \( \ln(x) \) and provides a way to express exponential growth deceleration.

• The natural logarithm is the inverse operation of exponentiation with the base \( e \), where \( e \approx 2.71828 \), a fundamental mathematical constant.
• In our demand function \( x = \ln \frac{1000}{p} \), \( \ln \) helps express the relationship logarithmically, simplifying the analysis of proportional changes.
• This logarithmic expression of demand function highlights how price changes logarithmically impact the demand, rather than linearly.