The derivative of `f(x) = (2x + 1)^(3) ` is

Options

A. ` 3(2x + 1)^(2)`
B. `6(2x + 1)^(2)`
C. `3(2x + 1)`
D. None of these

Explanation

To find the derivative of the function \( f(x) = (2x + 1)^3 \), we will use the chain rule of differentiation. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
Let \( u = 2x + 1 \). Then, we can rewrite the function as \( f(x) = u^3 \).

2. **Differentiate the outer function**:
The derivative of \( u^3 \) with respect to \( u \) is given by:
\[ \frac{d}{du}(u^3) = 3u^2 \]

3. **Differentiate the inner function**:
Now, we need to differentiate \( u = 2x + 1 \) with respect to \( x \):
\[ \frac{du}{dx} = 2 \]

4. **Apply the chain rule**:
According to the chain rule, the derivative of \( f(x) \) with respect to \( x \) is:
\[ \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{df}{dx} = 3u^2 \cdot 2 \]

5. **Substitute back the value of \( u \)**:
Now, substitute \( u = 2x + 1 \) back into the equation:
\[ \frac{df}{dx} = 3(2x + 1)^2 \cdot 2 \]

6. **Simplify the expression**:
Finally, simplifying gives:
\[ \frac{df}{dx} = 6(2x + 1)^2 \]

### Final Answer:
The derivative of \( f(x) = (2x + 1)^3 \) is:
\[ f'(x) = 6(2x + 1)^2 \]