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

Answer

To find the derivative of the function \( y = 2e^{x^2} \), we will use the chain rule. Here’s a step-by-step solution:

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

### Step 2: Use substitution
Let \( u = x^2 \). Then, we can rewrite the function in terms of \( u \):
\[ y = 2e^u \]

### Step 3: Differentiate \( y \) with respect to \( u \)
Now, we differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = 2e^u \]

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

### Step 5: Apply the chain rule
Using the chain rule, we find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = (2e^u) \cdot (2x) \]

### Step 6: Substitute back for \( u \)
Now, we substitute back \( u = x^2 \):
\[ \frac{dy}{dx} = 2e^{x^2} \cdot 2x = 4xe^{x^2} \]

### Final Answer
Thus, the derivative of \( y = 2e^{x^2} \) is:
\[ \frac{dy}{dx} = 4xe^{x^2} \]

---