The second derivative if `y =e^(2x^2)` is

Options

A. `4e^(2x^2)(4x^2+3)`
B. `4e^(2x^2)(4x^2-1)`
C. `4e^(2x^2)(4x^2+1)`
D. `e^(2x^2)(4x^2+1)`

Answer

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

### Step 1: Find the first derivative \( \frac{dy}{dx} \)

Given \( y = e^{2x^2} \), we will use the chain rule to differentiate.

Using the chain rule:
\[ \frac{dy}{dx} = \frac{d}{dx} e^{u} \cdot \frac{du}{dx} \]
where \( u = 2x^2 \).

Now, we differentiate \( e^{u} \):
\[ \frac{d}{dx} e^{u} = e^{u} \]
Next, we differentiate \( u = 2x^2 \):
\[ \frac{du}{dx} = 4x \]

Now, substituting back, we get:
\[ \frac{dy}{dx} = e^{2x^2} \cdot 4x = 4x e^{2x^2} \]

### Step 2: Find the second derivative \( \frac{d^2y}{dx^2} \)

Now we will differentiate \( \frac{dy}{dx} = 4x e^{2x^2} \) again using the product rule:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(4x) \cdot e^{2x^2} + 4x \cdot \frac{d}{dx}(e^{2x^2}) \]

Calculating the first part:
\[ \frac{d}{dx}(4x) = 4 \]
So,
\[ 4 \cdot e^{2x^2} \]

Now for the second part, we already found \( \frac{d}{dx}(e^{2x^2}) = 4x e^{2x^2} \):
\[ 4x \cdot 4x e^{2x^2} = 16x^2 e^{2x^2} \]

Combining both parts:
\[ \frac{d^2y}{dx^2} = 4 e^{2x^2} + 16x^2 e^{2x^2} \]
Factoring out \( e^{2x^2} \):
\[ \frac{d^2y}{dx^2} = e^{2x^2}(4 + 16x^2) \]

### Final Answer:
\[ \frac{d^2y}{dx^2} = e^{2x^2}(4 + 16x^2) \]

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