Suppose the demand for a particular commodity follows the equation:
where is the price and is the quantity sold. If the price is

increased from 6 to 6.5, what is the approximate fall in sales?

It seems you're referring to a demand function, where \( p \) is the price and \( q \) is the quantity sold. However, the demand equation is not provided in your query.

To solve this problem, we typically assume that demand functions are decreasing in price, meaning that as the price increases, the quantity sold decreases. One common form for a demand function is linear:

\[
q = a - bp
\]

where \( a \) and \( b \) are constants.

Alternatively, if the demand function is not linear but elastic, we may also work with a logarithmic or power function depending on the context.

Since you mentioned a price change from 6 to 6.5, let's assume we have a differentiable demand function \( q = f(p) \), and we are trying to approximate the fall in sales (or the change in quantity \( dq \)) as \( p \) increases from 6 to 6.5.

### Approach:

1. The approximate fall in sales can be estimated using the derivative of the demand function:
   \[
   dq \approx \frac{dq}{dp} \cdot \Delta p
   \]
   where \( \frac{dq}{dp} \) is the rate of change of quantity with respect to price (the derivative), and \( \Delta p \) is the change in price.

2. If the demand equation or elasticity is known, we can differentiate the function and plug in the values to estimate the fall in sales.

Please provide the form of the demand function or more details, so I can give you a precise answer!

Do you need help with the form of the demand function? Let me know!

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Here are 5 related questions for further understanding:
1. How do we derive the demand function if it's not provided?
2. What is price elasticity of demand and how does it affect changes in sales?
3. How can the derivative of a demand function help in forecasting sales?
4. What happens to sales if the demand function is nonlinear, such as quadratic or exponential?
5. How do changes in consumer behavior influence the demand curve?

**Tip**: The derivative of a demand function helps to understand the sensitivity of sales to price changes, known as **price elasticity**.

Suppose the demand for a particular commodity follows the equation: where is the price and is the quantity sold. If the price is increased from 6 to 6.5, what is the approximate fall in sales?

Learn how to estimate the fall in sales when the price of a commodity increases from 6 to 6.5 using calculus. This example explores differentiation of a demand function and the application of price elasticity of demand. Suitable for advanced high school or undergraduate economics students.

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