Question

Let f be a differentiable function with f(2) = 4. The function g is differentiable and g(x) = f⁻¹. What is the value of g'(4)?

Answer 1

To find the value of g'(4), we need to find the derivative of the function g(x) = f⁻¹ using the inverse function theorem.

Step-by-step explanation:

To find the value of g'(4), we need to find the derivative of the function g(x) = f⁻¹. Since g(x) is the inverse of f(x), we can use the inverse function theorem to find the derivative:

g'(x) = 1/f'(f⁻¹(x))

Since f(2) = 4, we know that f⁻¹(4) = 2. So, we can substitute x = 4 into the equation:

g'(4) = 1/f'(2)

Since we don't have the specific derivative of f(x), we can't determine the exact value of g'(4). However, we can say that g'(4) is equal to 1/f'(2).