Let f(x) be a bijective differentiable function and h(x) be its inverse function. It is given that f(1) = 2 , f(3) = 9 , f^ (1) = 3 , and f^ (3) = 1 2 . If a function g(x) is defined as g(x) = (h(x))^2 h(x^3 + 1) , then the value of g^ (2) is equal to:

Step-by-step solution

Since h(x) is the inverse of f(x) , we have f(h(x)) = x . From the given values: f(1) = 2 h(2) = 1 f(3) = 9 h(9) = 3 Using the derivative of an inverse function, h^ (x) = 1 f^ (h(x)) . At x = 2 : h^ (2) = 1 f^ (h(2)) = 1 f^ (1) = 1 3 At x = 9 : h^ (9) = 1 f^ (h(9)) = 1 f^ (3) = 1 1/2 = 2 Now, differentiate g(x) = (h(x))^2 h(x^3 + 1) using the product rule and chain rule: g^ (x) = [ 2h(x)h^ (x) ] h(x^3 + 1) + (h(x))^2 [ h^ (x^3 + 1) 3x^2 ] Substitute x = 2 into the derivative: g^ (2) = 2 h(2) h^ (2) h(9) + (h(2))^2