If `y= sin 7x + cos 5x + e^x` then `dy/dx`

Answer

To find the derivative \( \frac{dy}{dx} \) of the function \( y = \sin(7x) + \cos(5x) + e^x \), we will differentiate each term individually using the rules of differentiation.

### Step-by-step Solution:

1. **Identify the function**:
\[ y = \sin(7x) + \cos(5x) + e^x \]

2. **Differentiate \( \sin(7x) \)**:
- Using the chain rule, the derivative of \( \sin(mx) \) is \( m \cos(mx) \).
- Here, \( m = 7 \).
\[ \frac{d}{dx}[\sin(7x)] = 7 \cos(7x) \]

3. **Differentiate \( \cos(5x) \)**:
- Using the chain rule, the derivative of \( \cos(mx) \) is \( -m \sin(mx) \).
- Here, \( m = 5 \).
\[ \frac{d}{dx}[\cos(5x)] = -5 \sin(5x) \]

4. **Differentiate \( e^x \)**:
- The derivative of \( e^x \) is simply \( e^x \).
\[ \frac{d}{dx}[e^x] = e^x \]

5. **Combine the derivatives**:
- Now, we can combine the results from the previous steps:
\[ \frac{dy}{dx} = 7 \cos(7x) - 5 \sin(5x) + e^x \]

### Final Answer:
\[ \frac{dy}{dx} = 7 \cos(7x) - 5 \sin(5x) + e^x \]