Find dy/dx if y=`tan^2((pix^2)/2)`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \tan^2\left(\frac{\pi x^2}{2}\right)\), we will use the chain rule and the power rule of differentiation. Here’s a step-by-step solution:

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

### Step 2: Differentiate using the chain rule
We will differentiate \(y\) with respect to \(x\). Using the chain rule, we have:
\[ \frac{dy}{dx} = 2 \tan\left(\frac{\pi x^2}{2}\right) \cdot \frac{d}{dx}\left(\tan\left(\frac{\pi x^2}{2}\right)\right) \]

### Step 3: Differentiate \(\tan\left(\frac{\pi x^2}{2}\right)\)
The derivative of \(\tan(u)\) is \(\sec^2(u) \cdot \frac{du}{dx}\), where \(u = \frac{\pi x^2}{2}\). First, we find \(\frac{du}{dx}\):
\[ u = \frac{\pi x^2}{2} \implies \frac{du}{dx} = \frac{\pi}{2} \cdot 2x = \pi x \]
Now, we can differentiate \(\tan\left(\frac{\pi x^2}{2}\right)\):
\[ \frac{d}{dx}\left(\tan\left(\frac{\pi x^2}{2}\right)\right) = \sec^2\left(\frac{\pi x^2}{2}\right) \cdot \pi x \]

### Step 4: Substitute back into the derivative
Now we substitute back into our expression for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = 2 \tan\left(\frac{\pi x^2}{2}\right) \cdot \sec^2\left(\frac{\pi x^2}{2}\right) \cdot \pi x \]

### Step 5: Simplify the expression
Thus, we can write the final expression for \(\frac{dy}{dx}\) as:
\[ \frac{dy}{dx} = 2\pi x \tan\left(\frac{\pi x^2}{2}\right) \sec^2\left(\frac{\pi x^2}{2}\right) \]

### Final Result
\[ \frac{dy}{dx} = 2\pi x \tan\left(\frac{\pi x^2}{2}\right) \sec^2\left(\frac{\pi x^2}{2}\right) \]

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