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The given function \( f(x) \) is defined as:

\[
f(x) = -x^3 - 3x + 2
\]
and \( f(1) = -2 \). We are asked to find the value of \( g'(-2) \), where \( g \) is the inverse function of \( f \). 

To find \( g'(-2) \), we use the formula for the derivative of the inverse function:

\[
g'(y) = \frac{1}{f'(x)}
\]
where \( f(x) = y \). In this case, \( f(1) = -2 \), so we know \( x = 1 \) when \( y = -2 \). Thus, we need to compute \( f'(1) \), the derivative of \( f(x) \) evaluated at \( x = 1 \).

### Step 1: Find \( f'(x) \)
The derivative of \( f(x) \) is:

\[
f'(x) = \frac{d}{dx} \left( -x^3 - 3x + 2 \right) = -3x^2 - 3
\]

### Step 2: Evaluate \( f'(1) \)
Now, substitute \( x = 1 \) into the derivative:

\[
f'(1) = -3(1)^2 - 3 = -3 - 3 = -6
\]

### Step 3: Find \( g'(-2) \)
Finally, using the formula for the derivative of the inverse function:

\[
g'(-2) = \frac{1}{f'(1)} = \frac{1}{-6} = -\frac{1}{6}
\]

Thus, the value of \( g'(-2) \) is:

\[
g'(-2) = -\frac{1}{6}
\]

Would you like further details on any part of the solution?

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Here are 5 related questions for further exploration:

1. How do you find the inverse of a function algebraically?
2. What conditions must a function satisfy to have an inverse?
3. How do you interpret the meaning of the derivative of an inverse function?
4. What is the graphical relationship between a function and its inverse?
5. How would you compute the inverse of the function \( f(x) = -x^3 - 3x + 2 \) if possible?

**Tip:** The derivative of the inverse function tells us the rate of change of the original function’s output with respect to its input, but from the perspective of the inverse function.

The function f is defined by f(x) = -x^3 - 3x + 2 and f(1) = -2. If g is the inverse function of f, what is the value of g'(-2)?

Learn how to find the derivative of the inverse function g'(-2) for the function f(x) = -x^3 - 3x + 2. This solution explains the use of inverse function derivatives, key concepts in calculus, and the application of the inverse function theorem.

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