`(3x-5)(4x^(2) -3+e^(x))`

Answer

To differentiate the function \( y = (3x - 5)(4x^2 - 3 + e^x) \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u \) and \( v \), then the derivative of their product is given by:

\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step-by-Step Solution:

1. **Identify the Functions**:
Let \( u = 3x - 5 \) and \( v = 4x^2 - 3 + e^x \).

2. **Differentiate \( u \)**:
\[ \frac{du}{dx} = \frac{d}{dx}(3x - 5) = 3 \]

3. **Differentiate \( v \)**:
\[ \frac{dv}{dx} = \frac{d}{dx}(4x^2 - 3 + e^x) = 8x + 0 + e^x = 8x + e^x \]

4. **Apply the Product Rule**:
Now, using the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):
\[ \frac{dy}{dx} = (3x - 5)(8x + e^x) + (4x^2 - 3 + e^x)(3) \]

5. **Expand the Expression**:
- First term:
\[ (3x - 5)(8x + e^x) = 24x^2 + 3x e^x - 40x - 5e^x \]
- Second term:
\[ (4x^2 - 3 + e^x)(3) = 12x^2 - 9 + 3e^x \]

6. **Combine Like Terms**:
Now combine the two results:
\[ \frac{dy}{dx} = (24x^2 - 9) + (3x e^x - 5e^x + 3e^x) + (12x^2) \]
Combine the \( x^2 \) terms:
\[ \frac{dy}{dx} = (24x^2 + 12x^2 - 9) + (3x e^x - 2e^x) \]
\[ = 36x^2 - 9 + 3x e^x - 2e^x \]

### Final Answer:
\[ \frac{dy}{dx} = 36x^2 - 9 + 3x e^x - 2e^x \]