Q. \[ \frac{d}{dx}\left(\sin^2(x)\right) \]

We want to find the derivative of the function \(y=\sin^2(x)\).

Step 1: Rewrite the function in a form that makes the chain rule clear.

The expression \(\sin^2(x)\) means \(\big(\sin(x)\big)^2\). So we can write:

\[
y=\big(\sin(x)\big)^2
\]

Step 2: Identify the outer and inner functions (Chain Rule).

Think of \(y\) as a composition:

• Inner function: \(u=\sin(x)\)

• Outer function: \(y=u^2\)

Step 3: Differentiate using the Chain Rule.

The Chain Rule says:

\[
\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}
\]

Step 4: Compute each derivative separately.

First, differentiate the outer function \(y=u^2\) with respect to \(u\):

\[
\frac{dy}{du}=2u
\]

Next, differentiate the inner function \(u=\sin(x)\) with respect to \(x\):

\[
\frac{du}{dx}=\cos(x)
\]

Step 5: Substitute back \(u=\sin(x)\) and multiply.

Now combine them:

\[
\frac{dy}{dx}=2u\cdot \cos(x)
\]

Substitute \(u=\sin(x)\):

\[
\frac{dy}{dx}=2\sin(x)\cos(x)
\]

Final Answer.

\[
\frac{d}{dx}\left(\sin^2(x)\right)=2\sin(x)\cos(x)
\]