Differentiate `sin^(2) (2x+3)`

Answer

To differentiate the function \( y = \sin^2(2x + 3) \), we will use the chain rule. Here are the steps:

### Step 1: Identify the outer and inner functions
Let \( y = \sin^2(u) \) where \( u = 2x + 3 \).

### Step 2: Differentiate the outer function
Using the chain rule, the derivative of \( y = \sin^2(u) \) with respect to \( u \) is:
\[ \frac{dy}{du} = 2\sin(u)\cos(u) = \sin(2u) \quad \text{(using the double angle formula)} \]

### Step 3: Differentiate the inner function
Now, we differentiate the inner function \( u = 2x + 3 \):
\[ \frac{du}{dx} = 2 \]

### Step 4: Apply the chain rule
Now, we apply the chain rule to find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \sin(2u) \cdot 2 \]

### Step 5: Substitute back for \( u \)
Now substitute back \( u = 2x + 3 \):
\[ \frac{dy}{dx} = 2\sin(2(2x + 3)) = 2\sin(4x + 6) \]

### Final Answer
Thus, the derivative of \( y = \sin^2(2x + 3) \) is:
\[ \frac{dy}{dx} = 2\sin(4x + 6) \]

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