Question: 4. If \( f \) and \( g \) are functions differentiable everywhere, what is \( (f(x) \cdot…

4. If \( f \) and \( g \) are functions differentiable everywhere, what is \( (f(x) \cdot g(x))’ \)?

• \( f’(x) \cdot g(x) + f(x) \cdot g’(x) \)
• \( f’(x) \cdot g’(x) + f(x) \cdot g(x) \)
• \( f(x) \cdot g’(x) + f’(x) \cdot g’(x) \)

Solution

To find the derivative of the product of two functions \( f(x) \) and \( g(x) \), we use the Product Rule in differentiation. \[ \text{Product Rule: } (f(x) \cdot g(x))’ = f’(x) \cdot g(x) + f(x) \cdot g’(x) \] Applying the Product Rule to \( f(x) \cdot g(x) \): \[ (f(x) \cdot g(x))’ = f’(x) \cdot g(x) + f(x) \cdot g’(x) \] Therefore, the correct answer is: a) \( f’(x) \cdot g(x) + f(x) \cdot g’(x) \)