The second derivative of `y =sin^2 x` is :

Options

A. `cos 2x`
B. `2 sin 2x`
C. `2 sin x cos^2 x`
D. `2 cos 2x`

Answer

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

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

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

We can use the chain rule for differentiation. The chain rule states that if you have a function \( f(g(x)) \), then the derivative is given by:
\[ \frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx} \]

Here, let \( g(x) = \sin x \), so \( y = g^2 \). Thus, we have:
\[ \frac{dy}{dg} = 2g = 2\sin x \]
And,
\[ \frac{dg}{dx} = \cos x \]

Now applying the chain rule:
\[ \frac{dy}{dx} = 2\sin x \cdot \cos x \]

Using the double angle identity, we can simplify this:
\[ \frac{dy}{dx} = \sin(2x) \]

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

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

Using the chain rule again:
\[ \frac{d}{dx}(\sin(2x)) = \cos(2x) \cdot \frac{d}{dx}(2x) = 2\cos(2x) \]

### Final Result

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

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