`tan^2((pix^2)/2)`

Answer

To solve the problem of differentiating \( y = \tan^2\left(\frac{\pi x^2}{2}\right) \), we will use the chain rule and the derivative of the tangent function. Here’s a step-by-step solution:

### Step 1: Identify the function
Let:
\[ y = \tan^2\left(\frac{\pi x^2}{2}\right) \]

### Step 2: Apply the chain rule
To differentiate \( y \), we will use the chain rule. The derivative of \( \tan^2(u) \) is \( 2\tan(u) \cdot \sec^2(u) \cdot \frac{du}{dx} \), where \( u = \frac{\pi x^2}{2} \).

### Step 3: Differentiate the outer function
First, differentiate \( \tan^2(u) \):
\[ \frac{dy}{du} = 2\tan(u) \sec^2(u) \]

### Step 4: Differentiate the inner function
Now, differentiate \( u = \frac{\pi x^2}{2} \):
\[ \frac{du}{dx} = \frac{\pi}{2} \cdot 2x = \pi x \]

### Step 5: Combine the derivatives
Now, using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2\tan\left(\frac{\pi x^2}{2}\right) \sec^2\left(\frac{\pi x^2}{2}\right) \cdot \pi x \]

### Step 6: Write the final derivative
Thus, the derivative of \( y = \tan^2\left(\frac{\pi x^2}{2}\right) \) is:
\[ \frac{dy}{dx} = 2\pi x \tan\left(\frac{\pi x^2}{2}\right) \sec^2\left(\frac{\pi x^2}{2}\right) \]

### Summary
The final answer is:
\[ \frac{dy}{dx} = 2\pi x \tan\left(\frac{\pi x^2}{2}\right) \sec^2\left(\frac{\pi x^2}{2}\right) \]