To find the rate of change of the profit with respect to the number of units sold, we need to determine the derivative of the profit function.
Let's denote the profit function as P(x) = 590x - 0.3x^2.
By applying the power rule of derivation to each term of the function, we get the derivative P'(x), which represents the rate of change of the profit.
Here, the derivative of the first term, 590x, becomes 590 (as the derivative of x with respect to itself is 1). And the derivative of the second term, -0.3x^2, becomes -0.6x (using the power rule of derivation which states that the derivative of x^n = n*x^(n-1)).
So, P'(x) = 590 - 0.6x, which is the rate of change of the profit with respect to the number of units sold.
Now, let's calculate the rate of change when x equals 400 and 500.
(a) For x = 400 units, we substitute 400 for x in our derivative function P'(x):
P'(400) = 590 - 0.6*400 = 350.
Hence, when 400 units of the product are being sold per day, the profit is increasing at the rate of 350 units per day.
(b) For x = 500 units, we substitute 500 for x in our derivative function P'(x):
P'(500) = 590 - 0.6*500 = 290.
Hence, when 500 units of the product are being sold per day, the profit is increasing at the rate of 290 units per day.
Thus, as the number of units sold increases from 400 to 500, the rate of profit increase decreases from 350 to 290.