Let g(x) be the inverse of f(x) and `f'(x)=1/(1+x^(3))`.Find g'(x) in terms of g(x).

Answer

To find \( g'(x) \) in terms of \( g(x) \), where \( g(x) \) is the inverse of \( f(x) \) and \( f'(x) = \frac{1}{1 + x^3} \), we can follow these steps:

### Step 1: Understand the relationship between \( f \) and \( g \)
Since \( g(x) \) is the inverse of \( f(x) \), we have:
\[ f(g(x)) = x \]

### Step 2: Differentiate both sides
Differentiating both sides with respect to \( x \):
\[ \frac{d}{dx}[f(g(x))] = \frac{d}{dx}[x] \]
Using the chain rule on the left side, we get:
\[ f'(g(x)) \cdot g'(x) = 1 \]

### Step 3: Solve for \( g'(x) \)
Rearranging the equation to solve for \( g'(x) \):
\[ g'(x) = \frac{1}{f'(g(x))} \]

### Step 4: Substitute \( f'(x) \)
We know that \( f'(x) = \frac{1}{1 + x^3} \). Therefore, substituting \( g(x) \) into \( f' \):
\[ f'(g(x)) = \frac{1}{1 + (g(x))^3} \]

### Step 5: Final expression for \( g'(x) \)
Substituting this back into our expression for \( g'(x) \):
\[ g'(x) = \frac{1}{\frac{1}{1 + (g(x))^3}} = 1 + (g(x))^3 \]

Thus, the final result is:
\[ g'(x) = 1 + (g(x))^3 \]