Hello I need help with this excersise of derivative of a inverse function. The excersise says:

If f(x+2)=x^3+1 and g(x)=f(arctg x), find the derivative of the inverse function of g(x) and then calculate (g^-1 (9))'[the derivative of the inverse function evaluated in 9] .
I don't know which is the function f, they only give me f (x+2) and I don't understand how to get f
Please help

if

f(x+2) = x^3+1, then replace x with x-2 and you have
f(x-2+2) = f(x) = (x-2)^3 + 1
f^-1(x) = ∛(x-1) + 2

now just take the derivative of the inverse in the normal way.

(a) Find an angle between

0
and

that is coterminal with
−7π2
.
(b) Find an angle between

and
360°
that is coterminal with
586°

Sure, I can help you with that exercise!

To find the derivative of the inverse function of g(x), we first need to find the derivative of g(x) itself. Let's break it down step by step:

1. We're given that g(x) = f(arctan(x)). Here, arctan(x) represents the inverse tangent function.

2. We know that f(x+2) = x^3 + 1. Let's find f(x) by substituting x+2 in place of x in the equation: f(x) = (x+2)^3 + 1.

3. Now, we can substitute arctan(x) into the equation for f(x): f(arctan(x)) = (arctan(x)+2)^3 + 1.

4. Next, we find the derivative of g(x) by applying the chain rule.

The chain rule states that if we have a composition of two functions, such as f(g(x)), the derivative is given by the product of the derivative of the outer function (f'(g(x))) and the derivative of the inner function (g'(x)).

In our case, g(x) = f(arctan(x)), so the chain rule tells us that the derivative of g(x) is given by g'(x) = f'(arctan(x)) * (arctan(x))'.

5. Now, let's find the derivative of g(x) by differentiating each part:

a. The derivative of f(arctan(x)) with respect to arctan(x) is f'(arctan(x)). This is because the derivative of f(x) with respect to x is f'(x), and when we substitute arctan(x) in place of x, we get f'(arctan(x)).

b. The derivative of arctan(x) with respect to x is 1/(1+x^2). This is a well-known derivative.

6. Combining the derivatives from steps 5a and 5b, we get g'(x) = f'(arctan(x)) * (1/(1+x^2)).

Now that we have the derivative of g(x), we can calculate the derivative of the inverse function g^(-1)(x).

To calculate (g^(-1)(9))', we substitute 9 into the derivative expression we obtained in step 6, considering g^(-1)(x) as the input.

In summary, the steps are as follows:
1. Obtain the expression for f(x) by substituting x+2 in f(x+2).
2. Substitute arctan(x) into the expression for f(x) to get f(arctan(x)).
3. Apply the chain rule to differentiate g(x) = f(arctan(x)).
4. Simplify the derivative expression to get g'(x).
5. Substitute 9 into g'(x) to find (g^(-1)(9))'.

I hope this explanation helps you understand how to approach this exercise.