To find the derivative of the function \( y = [(x^2 + 2)(x^2 + 3)] \), you can use the product rule of differentiation, which states that if \( y = u \cdot v \), then \( y' = u'v + uv' \), where \( u' \) and \( v' \) are the derivatives of \( u \) and \( v \) with respect to \( x \) respectively.
Let's denote \( u = x^2 + 2 \) and \( v = x^2 + 3 \). Then, we have:
\( u' = \frac{d}{dx}(x^2 + 2) = 2x \) (derivative of \( x^2 + 2 \) with respect to \( x \))
\( v' = \frac{d}{dx}(x^2 + 3) = 2x \) (derivative of \( x^2 + 3 \) with respect to \( x \))
Now, applying the product rule:
\[ y' = u'v + uv' \]
\[ y' = (2x)(x^2 + 3) + (x^2 + 2)(2x) \]
\[ y' = 2x^3 + 6x + 2x^3 + 4x \]
\[ y' = 4x^3 + 10x \]
So, the derivative of the function \( y = [(x^2 + 2)(x^2 + 3)] \) with respect to \( x \) is \( y' = 4x^3 + 10x \).