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

Explanation

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

### Step 1: 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:
\[ 2x \cos(x^2) \]

### Step 2: Differentiate \( \sin^2(x) \)

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

### Step 3: Differentiate \( \sin^2(x^2) \)

Again, using the chain rule:
\[ \frac{d}{dx}[\sin^2(x^2)] = 2\sin(x^2) \cdot \frac{d}{dx}[\sin(x^2)] = 2\sin(x^2) \cdot \cos(x^2) \cdot \frac{d}{dx}[x^2] = 2\sin(x^2) \cdot \cos(x^2) \cdot 2x \]
So, the derivative of \( \sin^2(x^2) \) is:
\[ 4x \sin(x^2) \cos(x^2) \]

### Step 4: Combine all derivatives

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

### Final Answer

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

---