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

Explanation

To differentiate the function \( y = \sin(x^2) + \sin^2(x) + \sin^2(x^2) \), we will apply the chain rule and the product rule where necessary. Let's break it down step by step.

### Step 1: Identify the function
We have:
\[ y = \sin(x^2) + \sin^2(x) + \sin^2(x^2) \]

### Step 2: Differentiate each term
We will differentiate each term separately.

1. **Differentiate \( \sin(x^2) \)**:
- Using the chain rule:
\[ \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot \frac{d}{dx}(x^2) = \cos(x^2) \cdot 2x \]
So, the derivative of \( \sin(x^2) \) is \( 2x \cos(x^2) \).

2. **Differentiate \( \sin^2(x) \)**:
- Using the chain rule:
\[ \frac{d}{dx}[\sin^2(x)] = 2\sin(x) \cdot \frac{d}{dx}[\sin(x)] = 2\sin(x) \cdot \cos(x) \]
This simplifies to \( 2\sin(x)\cos(x) \), which can also be written as \( \sin(2x) \) using the double angle formula.

3. **Differentiate \( \sin^2(x^2) \)**:
- Again using the chain rule:
\[ \frac{d}{dx}[\sin^2(x^2)] = 2\sin(x^2) \cdot \frac{d}{dx}[\sin(x^2)] = 2\sin(x^2) \cdot \cos(x^2) \cdot \frac{d}{dx}(x^2) = 2\sin(x^2) \cdot \cos(x^2) \cdot 2x \]
This simplifies to \( 4x \sin(x^2) \cos(x^2) \).

### Step 3: Combine the derivatives
Now, we combine all the derivatives we calculated:
\[ \frac{dy}{dx} = 2x \cos(x^2) + 2\sin(x) \cos(x) + 4x \sin(x^2) \cos(x^2) \]

### Step 4: Final expression
Thus, the final expression for the derivative is:
\[ \frac{dy}{dx} = 2x \cos(x^2) + \sin(2x) + 4x \sin(x^2) \cos(x^2) \]

### Summary
The derivative of \( y = \sin(x^2) + \sin^2(x) + \sin^2(x^2) \) is:
\[ \frac{dy}{dx} = 2x \cos(x^2) + \sin(2x) + 4x \sin(x^2) \cos(x^2) \]