If `y=2sinx`, then `(d^2y)/(dx^2)` will be

Options

A. `2cos x`
B. `2sin x`
C. `- 2sin x`
D. `sin x + C`

Explanation

To solve the problem, we need to find the second derivative of the function \( y = 2 \sin x \). We will follow these steps:

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

Given:
\[ y = 2 \sin x \]

To find the first derivative, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(2 \sin x) \]

Using the constant multiple rule and the derivative of \( \sin x \):
\[ \frac{dy}{dx} = 2 \cdot \frac{d}{dx}(\sin x) = 2 \cos x \]

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

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

Again, using the constant multiple rule and the derivative of \( \cos x \):
\[ \frac{d^2y}{dx^2} = 2 \cdot \frac{d}{dx}(\cos x) = 2 \cdot (-\sin x) = -2 \sin x \]

### Final Answer

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

### Summary of Steps:
1. Differentiate \( y = 2 \sin x \) to find \( \frac{dy}{dx} = 2 \cos x \).
2. Differentiate \( \frac{dy}{dx} = 2 \cos x \) to find \( \frac{d^2y}{dx^2} = -2 \sin x \).