Differentiate `7x^3 + e^(4x)` w.r.t `x`

Answer

To differentiate the function \( y = 7x^3 + e^{4x} \) with respect to \( x \), we will apply the rules of differentiation step by step.

### Step-by-Step Solution:

1. **Identify the function to differentiate**:
\[ y = 7x^3 + e^{4x} \]

2. **Differentiate each term separately**:
- For the first term \( 7x^3 \):
- Use the power rule: If \( f(x) = ax^n \), then \( f'(x) = n \cdot ax^{n-1} \).
- Here, \( a = 7 \) and \( n = 3 \).
\[ \frac{d}{dx}(7x^3) = 3 \cdot 7x^{3-1} = 21x^2 \]

- For the second term \( e^{4x} \):
- Use the exponential rule: If \( f(x) = e^{mx} \), then \( f'(x) = m \cdot e^{mx} \).
- Here, \( m = 4 \).
\[ \frac{d}{dx}(e^{4x}) = 4 \cdot e^{4x} \]

3. **Combine the results**:
\[ \frac{dy}{dx} = \frac{d}{dx}(7x^3) + \frac{d}{dx}(e^{4x}) = 21x^2 + 4e^{4x} \]

4. **Final answer**:
\[ \frac{dy}{dx} = 21x^2 + 4e^{4x} \]