if f(x) = x^3 + e^x/2 and g[f(x)] = x, find the value of g' (1).

To solve for \( g'(1) \) given the functions \( f(x) = x^3 + \frac{e^x}{2} \) and \( g[f(x)] = x \), we proceed as follows:

### Step 1: Differentiate \( g[f(x)] = x \) with respect to \( x \)
Using the chain rule:
\[
\frac{d}{dx}[g(f(x))] = \frac{d}{dx}[x]
\]
\[
g'(f(x)) \cdot f'(x) = 1
\]

### Step 2: Solve for \( g'(f(x)) \)
\[
g'(f(x)) = \frac{1}{f'(x)}
\]

### Step 3: Evaluate \( f'(x) \)
The derivative of \( f(x) = x^3 + \frac{e^x}{2} \) is:
\[
f'(x) = 3x^2 + \frac{e^x}{2}
\]

### Step 4: Identify \( f(1) \)
Substitute \( x = 1 \) into \( f(x) \):
\[
f(1) = 1^3 + \frac{e^1}{2} = 1 + \frac{e}{2} = 1 + \frac{e}{2}
\]

### Step 5: Solve for \( g'(1) \)
Since \( g[f(1)] = 1 \), we know \( f(1) = 1 + \frac{e}{2} \). Using the formula for \( g'(f(x)) \):
\[
g'(1) = g'(f(1)) = \frac{1}{f'(1)}
\]

Substitute \( x = 1 \) into \( f'(x) \):
\[
f'(1) = 3(1^2) + \frac{e^1}{2} = 3 + \frac{e}{2}
\]

Thus:
\[
g'(1) = \frac{1}{f'(1)} = \frac{1}{3 + \frac{e}{2}}
\]

### Final Answer:
\[
g'(1) = \frac{2}{6 + e}
\]

Let me know if you'd like a detailed explanation of any step.  

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### Related Questions:
1. What is the general expression for \( g'(x) \) in terms of \( f(x) \)?
2. How does the chain rule work for nested functions?
3. What happens if \( f(x) \) changes to \( x^3 - \frac{e^x}{2} \)?
4. Can \( g'(x) \) ever equal zero? Why or why not?
5. How would this problem change if \( f'(x) \) had a critical point at \( x = 1 \)?

### Tip:
Always ensure that \( f(x) \) is invertible when solving such problems, as \( g[f(x)] = x \) implicitly assumes \( f(x) \) has an inverse.

If f(x) = x^3 + e^x/2 and g[f(x)] = x, find the value of g'(1).

Learn how to solve for g'(1) when f(x) = x^3 + e^x/2 and g[f(x)] = x using calculus. This step-by-step solution explains differentiation, the chain rule, and finding derivatives of inverse functions. Suitable for advanced high school or early college students.

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