Given \( g(x)=(3 x-1) e^{4 x^{2}} \), find \( g^{\prime}(x \) \( g^{\prime}(x)= \)

To find the derivative of the function \( g(x) = (3x - 1)e^{4x^{2}} \), we'll use the **product rule**. The product rule states that if you have a function \( h(x) = f(x) \cdot k(x) \), then its derivative is: \[ h'(x) = f'(x) \cdot k(x) + f(x) \cdot k'(x) \] Applying this to \( g(x) \): 1. **Identify \( f(x) \) and \( k(x) \):** - \( f(x) = 3x - 1 \) - \( k(x) = e^{4x^{2}} \) 2. **Find the derivatives \( f'(x) \) and \( k'(x) \):** - \( f'(x) = 3 \) - To find \( k'(x) \), use the chain rule: \[ k'(x) = \frac{d}{dx} \left( e^{4x^{2}} \right) = e^{4x^{2}} \cdot \frac{d}{dx}(4x^{2}) = e^{4x^{2}} \cdot 8x \] 3. **Apply the product rule:** \[ g'(x) = f'(x) \cdot k(x) + f(x) \cdot k'(x) = 3e^{4x^{2}} + (3x - 1)(8x)e^{4x^{2}} \] 4. **Factor out \( e^{4x^{2}} \) for simplification:** \[ g'(x) = e^{4x^{2}} \left(3 + (3x - 1)(8x)\right) \] 5. **Simplify the expression inside the parentheses:** \[ 3 + (24x^{2} - 8x) = 24x^{2} - 8x + 3 \] 6. **Write the final derivative:** \[ g'(x) = (24x^{2} - 8x + 3)e^{4x^{2}} \] **Final Answer:** \[ g^{\prime}(x) = \left(24\,x^{2} - 8\,x + 3\right) e^{4x^{2}} \]