Question

Find the rate of change of profit with respect to time, that is, find dP/dt given dt the following information. (Hint: Recall Profit = Revenue - Cost and use the chain rule) Revenue: R(x) = 6x Cost: C(x) = -x²+ 3x + 5 x =10, and dx dt = 5

Answer 1

To find the rate of change of profit with respect to time, use the chain rule. First, find dR/dt and dC/dt using the given revenue and cost functions. Finally, subtract dC/dt from dR/dt to find dP/dt.

Step-by-step explanation:

To find the rate of change of profit with respect to time, we can use the chain rule. The profit function is given by P(x) = R(x) - C(x), where R(x) is the revenue function and C(x) is the cost function. We are given R(x) = 6x and C(x) = -x² + 3x + 5. To find dP/dt, we need to find dR/dt and dC/dt first.

Using the chain rule, dR/dt = dR/dx * dx/dt = 6 * dx/dt = 6 * 5 = 30. Similarly, dC/dt = dC/dx * dx/dt = (-2x + 3) * dx/dt = (-2*10 + 3) * 5 = -95.

Finally, using the chain rule again, dP/dt = dR/dt - dC/dt = 30 - (-95) = 125.