λ = 0.2πE2 + E−
1
3− ln (1 − 2E−2)

To find the derivative of \(\lambda\) with respect to \(E\), we need to differentiate each term of the function:

\[
\lambda = 0.2\pi E^2 + E^{-1/3} - \ln(1 - 2E^{-2})
\]

Let's differentiate each term separately:

1. Differentiate \(0.2\pi E^2\):
\[
\frac{d}{dE}(0.2\pi E^2) = 0.2\pi \cdot 2E = 0.4\pi E
\]

2. Differentiate \(E^{-1/3}\):
\[
\frac{d}{dE}(E^{-1/3}) = -\frac{1}{3}E^{-4/3}
\]

3. Differentiate \(-\ln(1 - 2E^{-2})\):
\[
\frac{d}{dE}(-\ln(1 - 2E^{-2})) = -\frac{1}{1 - 2E^{-2}} \cdot \frac{d}{dE}(-2E^{-2})
\]
\[
\frac{d}{dE}(-2E^{-2}) = 4E^{-3}
\]
Thus,
\[
-\frac{1}{1 - 2E^{-2}} \cdot 4E^{-3} = \frac{4E^{-3}}{1 - 2E^{-2}}
\]

Combining these results, the derivative of \(\lambda\) with respect to \(E\) is:
\[
\frac{d\lambda}{dE} = 0.4\pi E - \frac{1}{3}E^{-4/3} + \frac{4E^{-3}}{1 - 2E^{-2}}
\]

Would you like further details or have any questions?

Here are five questions you might consider asking next:
1. How do you find the second derivative of this function?
2. What are the critical points of this function?
3. How do you integrate this function?
4. Can you explain how to simplify the derivative further?
5. What are some common mistakes when differentiating logarithmic functions?

Tip: When differentiating logarithmic functions, always remember to use the chain rule carefully to avoid errors.

Learn how to differentiate the function λ = 0.2πE² + E^(-1/3) - ln(1 - 2E^(-2)) step-by-step. This problem involves differentiation of polynomial, exponential, and logarithmic functions with application of the chain rule. Suitable for advanced high school or college-level students.

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