Questions: Suppose the demand for a certain item is given by D(p)=-3 p^2+5 p+3, where p represents the price of the item. Find D'(p), the rate of change of demand with respect to price. A. D'(p)=-3 p^2+5 B. D'(p)=-6 p^2+5 C. D'(p)=-6 p+5 D. D'(p)=-3 p+5

Suppose the demand for a certain item is given by D(p)=-3 p^2+5 p+3, where p represents the price of the item. Find D'(p), the rate of change of demand with respect to price.
A. D'(p)=-3 p^2+5
B. D'(p)=-6 p^2+5
C. D'(p)=-6 p+5
D. D'(p)=-3 p+5

Transcript text: Suppose the demand for a certain item is given by $\mathrm{D}(\mathrm{p})=-3 p^{2}+5 p+3$, where $p$ represents the price of the item. Find $\mathrm{D}^{\prime}(p)$, the rate of change of demand with respect to price. A. $D^{\prime}(p)=-3 p^{2}+5$ B. $D^{\prime}(p)=-6 p^{2}+5$ C. $D^{\prime}(p)=-6 p+5$ D. $D^{\prime}(p)=-3 p+5$

Solution

Solution Steps

Step 1: Identify the given function

The demand function is given by: \[ D(p) = -3p^2 + 5p + 3 \]

Step 2: Differentiate the function with respect to $ p $

To find the rate of change of demand with respect to price, we differentiate $ D(p) $ with respect to $ p $: \[ D'(p) = \frac{d}{dp} (-3p^2 + 5p + 3) \]

Step 3: Apply the power rule

Using the power rule for differentiation: \[ \frac{d}{dp} (-3p^2) = -6p \] \[ \frac{d}{dp} (5p) = 5 \] \[ \frac{d}{dp} (3) = 0 \]

Step 4: Combine the results

Adding the derivatives together: \[ D'(p) = -6p + 5 \]