The demand function for the Sicard wristwatch is given by the following function where d(x) (measured in units of a thousand) is the quantity demanded per week and x is the unit price in dollars.
d(x) =
50
0.01x2 + 1
(0 ≤ x ≤ 20)
(a) Find d'(x).
d'(x) =
(b) Find the following values. (Round your answers to three decimal places.)
d '(5) =
d '(10) =
d '(15) =
To find the derivative of the demand function, we need to use the power rule of differentiation. The power rule states that if f(x) = ax^n, then f'(x) = nax^(n-1).
(a) Applying the power rule to the given demand function, we have:
d(x) = (0.01x^2 + 1) / (0 ≤ x ≤ 20)
Using the power rule, we differentiate each term separately and obtain:
d'(x) = 0.02x
Therefore, d'(x) = 0.02x.
(b) Now we can calculate the values of d'(x) at specific points:
d'(5) = 0.02 * 5 = 0.1 (rounded to three decimal places)
d'(10) = 0.02 * 10 = 0.2 (rounded to three decimal places)
d'(15) = 0.02 * 15 = 0.3 (rounded to three decimal places)
Therefore, d'(5) = 0.1, d'(10) = 0.2, and d'(15) = 0.3.