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我的研究问题是：折射角 𝛼如何随水平距离 r1-r2 的变化而变化，现在如图算到这一步了，后面要怎么继续下去呢

The equation you've provided appears to be the derivative of a function related to the refraction angle \(\alpha\), with respect to the horizontal distance \(r_2\), likely involving parameters \(r_1\) and \(z_1\) (vertical distance). The formula is quite complex and involves differentiating a square root function with respect to \(r_2\).

To proceed further, I would recommend the following steps:

1. **Differentiate the remaining term**: The next step involves fully differentiating the term inside the brackets with respect to \(r_2\). You'll need to apply the **chain rule** and possibly the **quotient rule** because you have a complex expression involving \(r_2\) inside both a square root and a square.

2. **Simplify the expression**: Once you've differentiated, simplify the result as much as possible. This will help when you attempt to numerically calculate the refraction angle \(\alpha\) for different values of \(r_2\).

3. **Set up numerical methods**: The next step seems to suggest performing numerical simulations to compute the variation of the refraction angle \(\alpha\) for different values of \(r_2\), using the derived expression for \(\frac{df(r_2)}{dr_2}\).

4. **Apply boundary conditions**: If you have any boundary conditions or initial values for \(r_1\), \(r_2\), and \(z_1\), substitute those to make the equation more specific for your case.

5. **Simulate and plot**: Finally, use a mathematical tool like Python or MATLAB to simulate how \(\alpha\) changes as \(r_2\) varies and visualize it through plots.

Would you like help with differentiating the expression or setting up a numerical method for further analysis?

### Relative Questions:
1. What are the physical constraints or boundary conditions for \(r_1\), \(r_2\), and \(z_1\) in this model?
2. Is there a specific range of values for \(r_2\) that you're particularly interested in exploring?
3. How is the parameter 1.33 (likely the refractive index) influencing your overall model?
4. Would a numerical approach be preferred, or are you looking for an analytical solution?
5. Are you familiar with using Python or MATLAB for performing numerical simulations?

### Tip:
Using symbolic computation software like Wolfram Mathematica or SymPy in Python can help automate the differentiation and simplification of complex expressions.

我的研究问题是：折射角 𝛼 如何随水平距离 r1 - r2 的变化而变化，现在如图算到这一步了，后面要怎么继续下去呢

This solution explores how the refraction angle α changes with horizontal distance r2, involving a complex derivative that applies chain and quotient rules. Learn how to differentiate complex functions to analyze the refraction angle.

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