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Problem Overview:

The problem asks to find the functions $f \cdot g$, $f + g$, $f - g$, and $\frac{f}{g}$ for given functions $f(x)$ and $g(x)$, and then find their derivatives.

Given Functions:

1. First set:

   • $f(x) = x^3 + 2x^2$
   • $g(x) = 3x^2 - 1$

2. Second set:

   • $f(x) = \sqrt{1 + x^2}$
   • $g(x) = \frac{1}{1 + x^2}$

3. Third set:

   • $f(x) = x^2$
   • $g(x) = \frac{1}{x^2}$

Task:

For each pair of functions:

1. Compute the following:
   • $f \cdot g$
   • $f + g$
   • $f - g$
   • $\frac{f}{g}$
2. Compute the derivative of each of the above functions.

Would you like to proceed with solving all these tasks, or is there a specific part you would like to focus on first?