Let `f(x)a n dg(x)` be two function having finite nonzero third-order derivatives `f'''a n dg'''` for all `x in Rdot` If `f(x)g(x)=1` for all `x in R ,` then prove that `f'''(/)f^(prime)-g'''^(/)g^(prime)=3(f''^(/)f-g''^(/)g)dot`

Explanation

To prove the equation \( f'''(x)f'(x) - g'''(x)g'(x) = 3(f''(x)f(x) - g''(x)g(x)) \), we start with the given condition that \( f(x)g(x) = 1 \) for all \( x \in \mathbb{R} \).

### Step-by-step Solution:

1. **Differentiate the product**:
\[ f(x)g(x) = 1 \]
...