Lef f be the function defined by f(x)=x^3+x. If g(x) is the inverse of f(x), g(x)=f^-1(x) and g(2)=1, what is the value of g'(2)?
To find the value of g'(2), we need to use the inverse function theorem. The inverse function theorem states that if a function f(x) has an inverse function g(x), then the derivative of g(x) at a point x=a is equal to 1 divided by the derivative of f(x) at x=b, where b is the value of f(x) corresponding to x=a.
In this case, we are given g(2) = 1. Since g(x) is the inverse of f(x), this means that f(1) = 2.
To find the derivative of f(x), we can differentiate f(x) with respect to x:
f(x) = x^3 + x
f'(x) = 3x^2 + 1
Next, we need to find the value of x where f(x) = 2. From the given equation f(1) = 2, we know that x = 1 satisfies this condition.
Now, to find g'(2), we substitute x = 1 and b = 2 into the inverse function theorem formula:
g'(2) = 1 / f'(1)
= 1 / (3(1)^2 + 1)
= 1 / (3 + 1)
= 1 / 4
= 0.25
Therefore, the value of g'(2) is 0.25.