In the image, you're dealing with the product rule of differentiation applied using limits. The error highlighted seems to involve the incorrect term $g(x+h)$ when applying the product rule in its limit definition form.

The correct differentiation formula for the product of two functions $f(x)$ and $g(x)$ using limits is:

$(f(x)g(x))' = \lim_{h \to 0} \frac{f(x+h)g(x+h) - f(x)g(x)}{h}$

By expanding this and separating terms, we get the product rule:

$(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$

In the image:

• The part involving $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ is correct since it simplifies to $f'(x)$.
• The error comes from using $g(x+h)$ in a term that should have $g(x)$. Specifically, the term should contain $g(x)$ instead of $g(x+h)$, because when applying the product rule, one part differentiates $f(x)$ while holding $g(x)$ constant.

So the correct form for the expression should be:

$f'(x)g(x) + f(x)g'(x)$

To correct the error, replace the incorrect $g(x+h)$ with $g(x)$ in the first term. Therefore, the correct choice would be $g(x)$ from the options listed.

Would you like further details on this? Here are 5 questions to deepen the understanding:

1. Can you explain how the product rule applies to the combination of functions?
2. Why does the term involving $g(x+h)$ need to be replaced with $g(x)$?
3. How does the definition of the derivative help in deriving the product rule?
4. What would change if the functions were composed instead of multiplied?
5. Could this method be used for functions involving more complex products?

Tip: When applying the product rule, always check which function is being differentiated and ensure the other is treated as constant in that term.