If `y=e^(x)`, then `(d^(2)y)/(dx^(2)) = e^(x)`.

Answer

To prove that if \( y = e^x \), then \( \frac{d^2y}{dx^2} = e^x \), we will follow these steps:

### Step 1: Differentiate \( y \) with respect to \( x \)
Given:
\[ y = e^x \]
We need to find the first derivative \( \frac{dy}{dx} \).

Using the differentiation rule for the exponential function:
\[ \frac{dy}{dx} = e^x \]

### Step 2: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)
Now, we need to find the second derivative \( \frac{d^2y}{dx^2} \), which is the derivative of \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{dy}{dx} \right) = \frac{d}{dx} (e^x) \]

Again, using the differentiation rule for the exponential function:
\[ \frac{d^2y}{dx^2} = e^x \]

### Conclusion
We have shown that:
\[ \frac{d^2y}{dx^2} = e^x \]
Thus, the statement is proven to be true.

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