Please differentiate Li_2(1-x) with respect to x

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.

Please differentiate Li_2(1-x) with respect to x.

Learn how to differentiate the dilogarithm function Li_2(1-x) with respect to x. This solution includes key formulas for the dilogarithm function, such as the integral definition and its derivative, making it suitable for advanced calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation