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The rate of change in any function is a measure of how the output of the function varies as its input changes. In economics, the rate of change of a demand function reveals how the quantity demanded of a good or service changes in response to changes in its price.
In calculus, this is quantified by taking the derivative of the function with respect to the variable of interest, often price. The rate of change in demand with respect to price can help businesses and analysts understand consumer behavior and sensitivity to price adjustments.
In the supplied exercise, the rate of change reflects how quantity demanded, represented by x, varies as the price p changes. By calculating the derivative and evaluating it at a given price, we get a specific rate, which informs us by how much the demand is expected to decrease or increase with a rise or fall in price.

The derivative of the demand function with respect to price, often written as \( dx/dp \), is a mathematical representation of the sensitivity of demand relative to price changes. It is crucial for identifying optimal pricing strategies.
Considering the given demand function \( x=275\left(1-\frac{3p}{5p+1}\right) \), finding its derivative involves the application of rules of differentiation to handle the complexities that arise from the function's structure, which in this example includes a fraction. The derivative provides a snapshot of the demand's response at a particular price point.
In the step-by-step solution, we see the quotient rule is used to differentiate the function. After simplification, the derivative \( dx/dp \) is obtained, which can then be evaluated at any specific price to understand the demand's sensitivity to price at that point.

Differentiation is a fundamental operation in calculus used to find the rate at which a quantity changes with respect to another. When a function is a ratio of two other functions, as seen in many economic models including demand functions, the quotient rule is applied for differentiation.
For functions of the form \( u/v \), where both \( u \) and \( v \) are functions of \( p \), the quotient rule states that the derivative of \( u/v \) with respect \( p \) is \( (v(u' ) - u(v' ))/(v^2) \).
In the exercise we're examining, the demand function consists of a fraction that requires the quotient rule to correctly compute the derivative. The application of this rule followed by substantial simplification gives us the derivative function, which is then used to evaluate the rate of change for a specific value of the price.