Find `dy/dx` if `y=3x^3 + e^(7x) + 5`

Answer

To find the derivative \( \frac{dy}{dx} \) of the function \( y = 3x^3 + e^{7x} + 5 \), we will differentiate each term of the function step by step.

### Step-by-Step Solution:

1. **Identify the function**:
\[ y = 3x^3 + e^{7x} + 5 \]

2. **Differentiate the first term \( 3x^3 \)**:
- Using the power rule, which states that if \( y = x^n \), then \( \frac{dy}{dx} = n \cdot x^{n-1} \).
- Here, \( n = 3 \).
\[ \frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2 \]

3. **Differentiate the second term \( e^{7x} \)**:
- Using the chain rule for differentiation, which states that if \( y = e^{u} \) where \( u \) is a function of \( x \), then \( \frac{dy}{dx} = e^{u} \cdot \frac{du}{dx} \).
- Here, \( u = 7x \) and \( \frac{du}{dx} = 7 \).
\[ \frac{d}{dx}(e^{7x}) = e^{7x} \cdot 7 = 7e^{7x} \]

4. **Differentiate the constant term \( 5 \)**:
- The derivative of any constant is \( 0 \).
\[ \frac{d}{dx}(5) = 0 \]

5. **Combine the derivatives**:
- Now, we can combine the results from the differentiation of each term.
\[ \frac{dy}{dx} = 9x^2 + 7e^{7x} + 0 \]

6. **Final answer**:
\[ \frac{dy}{dx} = 9x^2 + 7e^{7x} \]

### Summary of the Solution:
The derivative of the function \( y = 3x^3 + e^{7x} + 5 \) is:
\[ \frac{dy}{dx} = 9x^2 + 7e^{7x} \]