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A demand equation represents the relationship between the quantity demanded of a good and its price. In mathematical terms, it connects the price (p) and the quantity demanded (x). For instance, in the given exercise, the demand equation is given implicitly as: ewlineewlineewline \[(p+4)*(x+3)=48\].ewlineewline Instead of representing the demand explicitly like \[p = f(x)\], this form shows a more interconnected relationship, indicating the combined impact on both price and demand. To work with such equations, we often need to use implicit differentiation to uncover how a change in quantity affects the price and vice versa. Understanding and manipulating these equations are crucial for analyzing market behavior.

The product rule is essential when differentiating products of two functions. If you have two functions, say \[u(x) \text{ and } v(x)\], their product's derivative is given by the formula:ewlineewline \[ \frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \].ewlineewline In the context of our demand equation \[(p+4)(x+3)=48\], we apply the product rule to differentiate terms like \[(p+4)\text{ and } (x+3) \]. Assuming \[(p+4) \text{ is } u(x) \] and \[(x+3) \text{ is } v(x) \], we can see that we need to treat \[(p)\] as a function of \[(x)\] to find \[ \frac{dp}{dx} \]. This approach leads to the formula: \[ p + x\frac{dp}{dx} \]. By adding the results from differentiating each term, we combine them to find the result of the differentiated equation.

A derivative represents how a function changes as its input changes. In simple terms, it's the rate at which one variable changes with respect to another. In this context, \[ \frac{dp}{dx} \] indicates how the price \[ (p) \] changes concerning quantity demanded \[ (x) \]. ewlineewline In our example, we expand and differentiate: \[(p+4)(x+3)=48\]. This requires implicit differentiation, a technique for finding derivatives when the relationship between the variables is given implicitly, not explicitly. Applying the product rule, we get: \[ p + x\frac{dp}{dx} + 3\frac{dp}{dx} + 4 = 0 \]. Combining and isolating: \[ (x+3)\frac{dp}{dx} = -(p + 4) \]. Thus, \[\frac{dp}{dx} = \frac{-(p + 4)}{x + 3}\]. This shows the derivative and how changes in demand affect pricing.