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

Options

A. cos 2x
B. 2 sin 2x
C. `2sinx 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:
\[ y = \sin^2 x \]

To differentiate \( y \) with respect to \( x \), we will use the chain rule. The chain rule states that if you have a function \( f(g(x)) \), then the derivative is given by \( f'(g(x)) \cdot g'(x) \).

Here, let \( u = \sin x \), then \( y = u^2 \).

Using the chain rule:
\[ \frac{dy}{dx} = 2u \cdot \frac{du}{dx} \]

Now, we know:
\[ \frac{du}{dx} = \cos x \]

Substituting back:
\[ \frac{dy}{dx} = 2 \sin x \cdot \cos x \]

Using the double angle identity:
\[ \sin 2x = 2 \sin x \cos x \]

Thus, we have:
\[ \frac{dy}{dx} = \sin 2x \]

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

Now, we need to differentiate \( \frac{dy}{dx} = \sin 2x \) again to find the second derivative.

Using the derivative of \( \sin x \):
\[ \frac{d}{dx}(\sin kx) = k \cos kx \]

In our case, \( k = 2 \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(\sin 2x) = 2 \cos 2x \]

### Final Result

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

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