Find the derivative of `y = e^(2x)`

Answer

To find the derivative of the function \( y = e^{2x} \), we can follow these steps:

### Step 1: Identify the function
The given function is:
\[ y = e^{2x} \]

### Step 2: Use the chain rule
To differentiate \( y \) with respect to \( x \), we will use the chain rule. The chain rule states that if you have a composite function \( y = e^{u} \) where \( u = 2x \), then:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

### Step 3: Differentiate \( y \) with respect to \( u \)
First, we differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = e^{u} \]

### Step 4: Differentiate \( u \) with respect to \( x \)
Next, we differentiate \( u = 2x \) with respect to \( x \):
\[ \frac{du}{dx} = 2 \]

### Step 5: Apply the chain rule
Now, we can apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = e^{u} \cdot 2 \]

### Step 6: Substitute back for \( u \)
We substitute back \( u = 2x \):
\[ \frac{dy}{dx} = 2e^{2x} \]

### Final Answer
Thus, the derivative of \( y = e^{2x} \) is:
\[ \frac{dy}{dx} = 2e^{2x} \]

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