The image you've provided appears to contain a mathematical problem related to differentiation. I will analyze the content and provide a solution.

From what I can see in the image:

1. The problem is asking for the derivative of a function $f(x) = \frac{x^2}{1 - e^x}$.

Let's proceed with finding the derivative of the given function.

Given:

$f(x) = \frac{x^2}{1 - e^x}$

To find the derivative, $f'(x)$, we will use the quotient rule: $\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}$

where $u = x^2$ and $v = 1 - e^x$.

Step-by-step Solution:

1. Find the derivatives of $u$ and $v$: $u' = 2x$ $v' = -e^x$

2. Apply the quotient rule: $f'(x) = \frac{(2x)(1 - e^x) - (x^2)(-e^x)}{(1 - e^x)^2}$

3. Simplify the numerator: $f'(x) = \frac{2x - 2xe^x + x^2e^x}{(1 - e^x)^2}$

So, the derivative of the function is:

$\boxed{f'(x) = \frac{2x - 2xe^x + x^2e^x}{(1 - e^x)^2}}$

Would you like more details on any of the steps?

Here are some related questions that might be useful:

1. How can we apply the chain rule to more complex functions?
2. What are some common mistakes to avoid when using the quotient rule?
3. How do we handle indeterminate forms in differentiation?
4. Can we use L'Hopital's rule to solve limits involving derivatives?
5. How is the product rule different from the quotient rule?

Tip: Always simplify the function as much as possible before differentiating to make the process easier.