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The product rule is a fundamental principle in calculus used to differentiate functions that are the product of two or more expressions. Imagine you have two functions, such as \( u(x) = (p-1) \) and \( v(x) = (x+5) \). When these functions are multiplied, their derivative is not simply the product of their individual derivatives. Instead, the product rule provides a way to find the correct derivative when two functions are multiplied.

In mathematical terms, if \( y = u \cdot v \), then the derivative \( \frac{dy}{dx} \) is calculated as:

• \( u'v + uv' \)

This means you take the derivative of the first function \( u \), multiply it by the second function \( v \), and add the product of the first function \( u \) and the derivative of the second function \( v \).

When applying the product rule to the demand equation \((p-1)(x+5)\), you end up differentiating each term appropriately. The process helps in finding how a change in one variable affects another in complex products.

Understanding derivative rules helps to make differentiation straightforward. There are several basic rules of differentiation that you consistently use.

• **Constant Rule:** The derivative of a constant is zero. If \( c \) is a constant, then \( \frac{d}{dx}[c] = 0 \).
• **Power Rule:** For any real number \( n \), \( \frac{d}{dx}[x^n] = nx^{n-1} \).
• **Sum Rule:** The derivative of a sum equals the sum of derivatives, \( \frac{d}{dx}[u + v] = \frac{du}{dx} + \frac{dv}{dx} \).
• **Product Rule:** As previously explained, use when differentiating a product of terms.
• **Chain Rule:** Useful for differentiating composite functions, \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \).

In the context of implicit differentiation, these rules help in tackling equations where one variable is a function of another. For example, in \( (p-1)(x+5) = 24 \), you're not given \( p \) as an explicit function of \( x \), so implicit differentiation and these rules let you find \( \frac{dp}{dx} \) more effectively.

In economics, a demand equation outlines the relationship between the quantity demanded of a good and its price. It considers various factors such as population segment, income level, or price changes.

The given demand equation \((p-1)(x+5) = 24\) is an implicit form that relates price \( p \) and quantity \( x \). Solving for \( \frac{dp}{dx} \), using implicit differentiation, helps to understand how demand \( x \) changes with respect to price \( p \). Applying implicit differentiation provides insights into elasticity and slope of the demand curve, which are vital in predicting consumer behavior.

In real-world applications, such equations help firms set prices or assess potential revenue from changes in market conditions. They provide a basis for further economic analysis by understanding the gradient of change between variables, like price and demand.