We are tasked with finding $f'(1)$ given the equation

$x f(x) g(x) = 2 f(g(x)) - 2,$

with the additional information that $g(1) = 1$ and $g'(1) = 3$. We need to use implicit differentiation with respect to $x$.

Step 1: Differentiate both sides of the equation

Using the product rule and chain rule, differentiate both sides of the equation with respect to $x$.

Left-hand side:

$\frac{d}{dx} \left( x f(x) g(x) \right) = f(x) g(x) + x \left( f'(x) g(x) + f(x) g'(x) \right).$

Right-hand side:

$\frac{d}{dx} \left( 2 f(g(x)) - 2 \right) = 2 f'(g(x)) g'(x).$

Step 2: Set up the equation after differentiation

Equating both sides, we have:

$f(x) g(x) + x \left( f'(x) g(x) + f(x) g'(x) \right) = 2 f'(g(x)) g'(x).$

Step 3: Substitute known values at $x = 1$

We are given $g(1) = 1$, $g'(1) = 3$, and we need to find $f'(1)$.

Substitute $x = 1$ into the differentiated equation:

$f(1) g(1) + 1 \left( f'(1) g(1) + f(1) g'(1) \right) = 2 f'(g(1)) g'(1).$

Using $g(1) = 1$, $g'(1) = 3$, and simplifying:

$f(1) \cdot 1 + \left( f'(1) \cdot 1 + f(1) \cdot 3 \right) = 2 f'(1) \cdot 3.$

This simplifies to:

$f(1) + f'(1) + 3 f(1) = 6 f'(1).$

Step 4: Solve for $f'(1)$

We are not directly given $f(1)$, but solving for $f'(1)$, we get:

$f(1) + f'(1) + 3 f(1) = 6 f'(1),$

or:

$4 f(1) + f'(1) = 6 f'(1),$

which simplifies to:

$4 f(1) = 5 f'(1).$

Thus,

$f'(1) = \frac{4 f(1)}{5}.$

Step 5: Use $f(1) = 1$ (implied assumption)

While not explicitly stated, based on the structure and reasoning of similar problems, it is often assumed that $f(1) = 1$. With this assumption:

$f'(1) = \frac{4 \times 1}{5} = \frac{4}{5} = 0.8.$

Thus, the value of $f'(1)$ is:

$\boxed{0.8}.$

Would you like to explore any specific part of this solution in more detail or ask questions about the process?

Here are 5 related questions to further your understanding:

1. How would the result change if $f(1)$ were a different value?
2. Can you differentiate both sides without using the product rule?
3. What would happen if $g'(1)$ were not 3 but another value?
4. Can this implicit differentiation approach be used in cases where $g(x)$ is not given explicitly?
5. How does the chain rule apply when differentiating $f(g(x))$?

Tip: Always verify all assumptions in a problem, especially if certain values like $f(1)$ are not explicitly given but might be deduced from context.