The relationship between the amount of money x that Cannon Precision Instruments spends on advertising and the company's total sales S(x) is given by the following function where x is measured in thousands of dollars. S(x) = −0.002x3 + 0.4x2 + 9x + 500 (0 ≤ x ≤ 200) Find the rate of change of the sales with respect to the amount of money spent on advertising.

Answer:S'(x)=-0.006x2+0.8x+9Step-by-step explanation:The rate of change of the sales S(x) with respect to the amount of money spent on advertising x is then the derivative of S(x) with respect to x. What we have to calculate is S'(x).We know that the derivative of [tex]ax^n[/tex] is [tex]nax^{n-1}[/tex] (where a is a constant), and the derivative of a sum is the sum of the derivatives, so we can calculate the derivative of each term independently:S'(x)=(−0.002x3 + 0.4x2 + 9x + 500)'=(−0.002)(3)x2 + (0.4)(2)x + 9Which means:S'(x)=-0.006x2+0.8x+9