If `y= 2x^6 + sin 3x` then `dy/dx`

Answer

To find the derivative of the function \( y = 2x^6 + \sin(3x) \), we will differentiate each term separately using the rules of differentiation.

### Step-by-Step Solution:

1. **Identify the function**:
We have \( y = 2x^6 + \sin(3x) \).

2. **Differentiate the first term \( 2x^6 \)**:
- We use the power rule of differentiation:
\[ \frac{d}{dx}(x^n) = n \cdot x^{n-1} \]
- Here, \( n = 6 \), so:
\[ \frac{d}{dx}(2x^6) = 2 \cdot 6 \cdot x^{6-1} = 12x^5 \]

3. **Differentiate the second term \( \sin(3x) \)**:
- We use the chain rule for differentiation:
\[ \frac{d}{dx}(\sin(mx)) = m \cdot \cos(mx) \]
- Here, \( m = 3 \), so:
\[ \frac{d}{dx}(\sin(3x)) = 3 \cdot \cos(3x) \]

4. **Combine the derivatives**:
- Now, we combine the results from the differentiation of both terms:
\[ \frac{dy}{dx} = 12x^5 + 3\cos(3x) \]

5. **Final answer**:
\[ \frac{dy}{dx} = 12x^5 + 3\cos(3x) \]