If `(d)/(dx)[f(x)]=(1)/(1+x^(2))," then: "(d)/(dx)[f(x^(3))]=`

Options

A. `(3x)/(1+x^(3))`
B. `(3x^(2))/(1+x^(6))`
C. `(-6x^(5))/((1+x^(6))^(2))`
D. `(-6x^(5))/(1+x^(6))`

Answer

To solve the problem, we need to differentiate \( f(x^3) \) given that \( \frac{d}{dx}[f(x)] = \frac{1}{1+x^2} \).

### Step-by-Step Solution:

1. **Identify the given information**:
We know that \( \frac{d}{dx}[f(x)] = \frac{1}{1+x^2} \).

2. **Use the chain rule for differentiation**:
We need to differentiate \( f(x^3) \). According to the chain rule, if \( y = f(g(x)) \), then:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
Here, \( g(x) = x^3 \).

3. **Differentiate \( g(x) \)**:
Calculate \( g'(x) \):
\[ g'(x) = \frac{d}{dx}[x^3] = 3x^2 \]

4. **Substitute \( g(x) \) into \( f'(x) \)**:
We need to find \( f'(g(x)) = f'(x^3) \). From the given information, we have:
\[ f'(x) = \frac{1}{1+x^2} \]
Therefore:
\[ f'(x^3) = \frac{1}{1+(x^3)^2} = \frac{1}{1+x^6} \]

5. **Combine the results**:
Now substitute \( f'(x^3) \) and \( g'(x) \) into the chain rule formula:
\[ \frac{d}{dx}[f(x^3)] = f'(x^3) \cdot g'(x) = \frac{1}{1+x^6} \cdot 3x^2 \]

6. **Final result**:
Thus, the derivative \( \frac{d}{dx}[f(x^3)] \) is:
\[ \frac{3x^2}{1+x^6} \]

### Final Answer:
\[ \frac{d}{dx}[f(x^3)] = \frac{3x^2}{1+x^6} \]