Let f be a differentiable function defined for all `x in R` such that `f(x^(3))=x^(5)` fol all `x in R,xne0`. Then the value of `f'(8)`, is

Options

A. 20
B. `(20)/(3)`
C. `(5)/(3)`
D. none of these

Explanation

To solve the problem, we need to find the derivative \( f'(8) \) given the equation \( f(x^3) = x^5 \).

### Step-by-step Solution:

1. **Differentiate both sides with respect to \( x \)**:
\[ \frac{d}{dx}[f(x^3)] = \frac{d}{dx}[x^5] \]

2. **Apply the chain rule on the left side**:
\[ f'(x^3) \cdot \frac{d}{dx}(x^3) = 5x^4 \]
Here, \( \frac{d}{dx}(x^3) = 3x^2 \).

3. **Substituting the derivative**:
\[ f'(x^3) \cdot 3x^2 = 5x^4 \]

4. **Solve for \( f'(x^3) \)**:
\[ f'(x^3) = \frac{5x^4}{3x^2} = \frac{5}{3} x^2 \]

5. **Now, we need to find \( f'(8) \)**. To do this, we need to find \( x \) such that \( x^3 = 8 \):
\[ x = 2 \quad (\text{since } 2^3 = 8) \]

6. **Substituting \( x = 2 \) into \( f'(x^3) \)**:
\[ f'(8) = f'(2^3) = \frac{5}{3}(2^2) = \frac{5}{3} \cdot 4 = \frac{20}{3} \]

### Final Answer:
\[ f'(8) = \frac{20}{3} \]