Final answer:
To find the derivative of the inverse function f-1(a) at a given point, first determine f'(x), then find x corresponding to the given a, evaluate f'(x) at this x, and take the reciprocal to get (f-1)'(a). The positive root is used for x because negative concentrations are not physical.Correct option is b.
Step-by-step explanation:
To find the derivative of f-1(a) when a = 2, we need to apply the formula for the derivative of an inverse function. Specifically, we'll use the fact that the derivative of an inverse function f-1(a) at a point a is given by 1 over the derivative of the original function f'(x) evaluated at the point x = f-1(a).
Factoring and solving f(x) = 2, we find that x has two possible values (ignoring the negative root since it's not physically meaningful in this context). We're only considering the positive root which is x = 7.2 x 10-2. Next, we evaluate the derivative at this value, f'(7.2 x 10-2).
Plugging in x, we get f'(7.2 x 10-2) which calculates to a specific positive value. Finally, we find the reciprocal of this derivative to get the derivative of the inverse function at a = 2. The correct answer yields one of the given options, ensuring to match the question's function rather than an unrelated equation provided in the reference.