Final answer:
To find (f^(-1))^'(1), we need to find f^(-1)(1) and f'(f^(-1)(1)). Using the formula (f^(-1))^'(x) = 1 / f'(f^(-1)(x)), we can substitute these values to find the derivative of the inverse function.
Step-by-step explanation:
To find the derivative of the inverse function, we can use the formula (f^(-1))^'(x) = 1 / f'(f^(-1)(x)). In this case, we want to find (f^(-1))^'(1). So we need to find f^(-1)(1) and f'(f^(-1)(1)).
First, we find f^(-1)(1) by setting 2x + cos(x) = 1 and solving for x. This gives us x = f^(-1)(1) = 0.450183.
Next, we find f'(f^(-1)(1)). The derivative of f(x) = 2x + cos(x) is f'(x) = 2 - sin(x). Plugging in x = f^(-1)(1) = 0.450183, we get f'(f^(-1)(1)) = 2 - sin(0.450183) = 1.71634.
Finally, we can substitute these values into the formula (f^(-1))^'(x) = 1 / f'(f^(-1)(x)) to find (f^(-1))^'(1) = 1 / f'(f^(-1)(1)) = 1 / 1.71634 = 0.582845.'