if `f ( x ) = 2x e^ (2x)` find f ' ( x )

Answer

To find the derivative \( f'(x) \) of the function \( f(x) = 2x e^{2x} \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

\[ (uv)' = u'v + uv' \]

### Step-by-Step Solution

1. **Identify the functions**:
Let \( u = 2x \) and \( v = e^{2x} \).

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) is:
\[ u' = \frac{d}{dx}(2x) = 2 \]
- The derivative of \( v \) is:
\[ v' = \frac{d}{dx}(e^{2x}) = e^{2x} \cdot \frac{d}{dx}(2x) = e^{2x} \cdot 2 = 2e^{2x} \]
(Here, we used the chain rule for differentiating \( e^{2x} \)).

3. **Apply the product rule**:
Now, apply the product rule:
\[ f'(x) = u'v + uv' \]
Substituting the values we found:
\[ f'(x) = (2)(e^{2x}) + (2x)(2e^{2x}) \]

4. **Simplify the expression**:
\[ f'(x) = 2e^{2x} + 4xe^{2x} \]
We can factor out \( e^{2x} \):
\[ f'(x) = e^{2x}(2 + 4x) \]

### Final Answer
Thus, the derivative of the function is:
\[ f'(x) = e^{2x}(4x + 2) \]