Find the second order derivative if `y= e^(2x)`

Answer

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

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

Given:
\[ y = e^{2x} \]

To differentiate \( y \) with respect to \( x \), we apply the chain rule. The derivative of \( e^{u} \) with respect to \( x \) is \( e^{u} \cdot \frac{du}{dx} \), where \( u = 2x \).

Calculating:
\[ \frac{dy}{dx} = e^{2x} \cdot \frac{d(2x)}{dx} = e^{2x} \cdot 2 = 2e^{2x} \]

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

Now, we differentiate \( \frac{dy}{dx} = 2e^{2x} \) with respect to \( x \).

Calculating:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(2e^{2x}) = 2 \cdot e^{2x} \cdot \frac{d(2x)}{dx} = 2 \cdot e^{2x} \cdot 2 = 4e^{2x} \]

### Final Result

Thus, the second-order derivative of \( y = e^{2x} \) is:
\[ \frac{d^2y}{dx^2} = 4e^{2x} \]

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