find \( \frac{d y}{d x} \) \( \tan ^{2}\left(3 x^{2}\right)=y \)

1. We start with the given function: \[ y = \tan^2(3x^2). \] 2. To differentiate \( y \) with respect to \( x \), we use the chain rule. Write: \[ y = \left[\tan(3x^2)\right]^2. \] Differentiating, we have: \[ \frac{dy}{dx} = 2 \tan(3x^2) \cdot \frac{d}{dx}\left[\tan(3x^2)\right]. \] 3. Next, differentiate \( \tan(3x^2) \) using the chain rule. We know: \[ \frac{d}{dx}\left[\tan(u)\right] = \sec^2(u) \cdot \frac{du}{dx}, \] where in this case \( u = 3x^2 \). 4. Compute the derivative of \( u = 3x^2 \): \[ \frac{du}{dx} = 6x. \] Therefore: \[ \frac{d}{dx}\left[\tan(3x^2)\right] = \sec^2(3x^2) \cdot 6x. \] 5. Substitute back into the derivative of \( y \): \[ \frac{dy}{dx} = 2 \tan(3x^2) \cdot \left[\sec^2(3x^2) \cdot 6x\right]. \] Simplifying, we obtain: \[ \frac{dy}{dx} = 12x \tan(3x^2) \sec^2(3x^2). \] Thus, the derivative is: \[ \frac{dy}{dx} = 12x \tan(3x^2)\sec^2(3x^2). \]