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 of differentiation step by step.

### Step-by-step Solution:

1. **Identify the function**:
\[ y = \sin(x^2) + \sin^2(x) + \sin^2(x^2) \]

2. **Differentiate each term**:
We will differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}[\sin(x^2)] + \frac{d}{dx}[\sin^2(x)] + \frac{d}{dx}[\sin^2(x^2)] \]

3. **Differentiate \( \sin(x^2) \)**:
Using the chain rule:
\[ \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot \frac{d}{dx}[x^2] = \cos(x^2) \cdot 2x \]

4. **Differentiate \( \sin^2(x) \)**:
Using the chain rule:
\[ \frac{d}{dx}[\sin^2(x)] = 2\sin(x) \cdot \frac{d}{dx}[\sin(x)] = 2\sin(x) \cdot \cos(x) \]

5. **Differentiate \( \sin^2(x^2) \)**:
Using the chain rule:
\[ \frac{d}{dx}[\sin^2(x^2)] = 2\sin(x^2) \cdot \frac{d}{dx}[\sin(x^2)] = 2\sin(x^2) \cdot \cos(x^2) \cdot \frac{d}{dx}[x^2] = 2\sin(x^2) \cdot \cos(x^2) \cdot 2x \]

6. **Combine all derivatives**:
Now we can combine all the derivatives we calculated:
\[ \frac{dy}{dx} = 2x\cos(x^2) + 2\sin(x)\cos(x) + 4x\sin(x^2)\cos(x^2) \]

7. **Final expression**:
Thus, the derivative of the function is:
\[ \frac{dy}{dx} = 2x\cos(x^2) + 2\sin(x)\cos(x) + 4x\sin(x^2)\cos(x^2) \]