If `y=Ae^(5x)`, then `(d^(2)y)/(dx^(2))` is equal to

Options

A. 25 y
B. 5 y
C. `-25y`
D. 15 y

Answer

To find the second derivative of the function \( y = Ae^{5x} \), we will follow these steps:

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

Given:
\[ y = Ae^{5x} \]

To differentiate \( y \) with respect to \( x \), we use the chain rule. The derivative of \( e^{kx} \) is \( ke^{kx} \).

Thus, we have:
\[ \frac{dy}{dx} = A \cdot \frac{d}{dx}(e^{5x}) = A \cdot 5e^{5x} = 5Ae^{5x} \]

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

Now, we differentiate \( \frac{dy}{dx} \) to find the second derivative:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(5Ae^{5x}) \]

Again, applying the chain rule:
\[ \frac{d^2y}{dx^2} = 5A \cdot \frac{d}{dx}(e^{5x}) = 5A \cdot 5e^{5x} = 25Ae^{5x} \]

### Step 3: Express \( \frac{d^2y}{dx^2} \) in terms of \( y \)

Since \( y = Ae^{5x} \), we can substitute \( y \) into our expression for the second derivative:
\[ \frac{d^2y}{dx^2} = 25y \]

### Final Answer
Thus, the second derivative \( \frac{d^2y}{dx^2} \) is:
\[ \frac{d^2y}{dx^2} = 25y \]

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