Differentiate `x^(3)+3^(3)+3^(x)` with respect to x.

Answer

To differentiate the function \( f(x) = x^3 + 3^3 + 3^x \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function
The function we need to differentiate is:
\[ f(x) = x^3 + 3^3 + 3^x \]
Here, \( 3^3 \) is a constant.

### Step 2: Differentiate each term
We will differentiate each term in the function separately.

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

2. **Differentiate \( 3^3 \)**:
Since \( 3^3 \) is a constant (equal to 27), its derivative is:
\[ \frac{d}{dx}(3^3) = 0 \]

3. **Differentiate \( 3^x \)**:
For the term \( 3^x \), we use the exponential differentiation rule, which states that if \( f(x) = a^x \), then \( f'(x) = a^x \ln(a) \):
\[ \frac{d}{dx}(3^x) = 3^x \ln(3) \]

### Step 3: Combine the derivatives
Now, we combine the derivatives of each term:
\[ f'(x) = 3x^2 + 0 + 3^x \ln(3) \]
Thus, we can simplify this to:
\[ f'(x) = 3x^2 + 3^x \ln(3) \]

### Final Answer
The derivative of the function \( f(x) = x^3 + 3^3 + 3^x \) with respect to \( x \) is:
\[ f'(x) = 3x^2 + 3^x \ln(3) \]