If `y=x^(x)+x^(3)+3^(x)+3^(3)," fin d"(dy)/(dx)`

Answer

To find the derivative of the function \( y = x^x + x^3 + 3^x + 3^3 \), we will differentiate each term separately.

### Step-by-Step Solution:

1. **Differentiate \( x^x \)**:
- The derivative of \( x^x \) can be found using the formula:
\[ \frac{d}{dx}(x^x) = x^x \left(1 + \log x\right) \]

2. **Differentiate \( x^3 \)**:
- The derivative of \( x^3 \) is straightforward:
\[ \frac{d}{dx}(x^3) = 3x^2 \]

3. **Differentiate \( 3^x \)**:
- The derivative of \( 3^x \) is given by:
\[ \frac{d}{dx}(3^x) = 3^x \log 3 \]

4. **Differentiate \( 3^3 \)**:
- Since \( 3^3 \) is a constant, its derivative is:
\[ \frac{d}{dx}(3^3) = 0 \]

5. **Combine all the derivatives**:
- Now, we can combine all the derivatives to find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^x) + \frac{d}{dx}(x^3) + \frac{d}{dx}(3^x) + \frac{d}{dx}(3^3) \]
\[ \frac{dy}{dx} = x^x(1 + \log x) + 3x^2 + 3^x \log 3 + 0 \]

6. **Final Result**:
- Therefore, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = x^x(1 + \log x) + 3x^2 + 3^x \log 3 \]