To differentiate $\text{Li}_2(1-x)$ with respect to $x$, we use the fact that $\text{Li}_2(z)$ is the dilogarithm function, which is defined as:

$\text{Li}_2(z) = -\int_0^z \frac{\ln(1-t)}{t} dt$

A key property of the dilogarithm function is its derivative:

$\frac{d}{dz} \text{Li}_2(z) = -\frac{\ln(1-z)}{z}$

Now, apply this to $\text{Li}_2(1-x)$. First, differentiate with respect to $x$:

$\frac{d}{dx} \text{Li}_2(1-x) = -\frac{\ln(x)}{1-x}$

Thus, the derivative of $\text{Li}_2(1-x)$ with respect to $x$ is:

$\boxed{\frac{d}{dx} \text{Li}_2(1-x) = -\frac{\ln(x)}{1-x}}$

Would you like further details on dilogarithms or the derivation?

Here are 5 related questions to explore this further:

1. What is the dilogarithm function and how is it used in mathematics?
2. How can we differentiate higher-order polylogarithms?
3. What is the relation between the dilogarithm function and the Riemann zeta function?
4. How can you numerically approximate $\text{Li}_2(z)$?
5. Are there any special values for $\text{Li}_2(z)$ that are useful in physics or number theory?

Tip: The polylogarithm functions, including the dilogarithm, frequently appear in quantum field theory and number theory, especially when dealing with sums of series or integrals.