Okay, I need to create a Korean wiki-style markdown document about the Chain Rule in calculus. Let's start by understanding the user's requirements. The main category is Technology, then Math, specifically Calculus, with the title being the Chain Rule. The document should have a clear H1 title, an introduction, several detailed sections using H2 and H3, and possibly references. The content must be accurate, objective, with adequate depth and easy to read. Also, proper markdown formatting with lists, tables, and code blocks where necessary.

First, the title. It should be in H1, so "# 체인 규칙" makes sense. The introduction should explain what the Chain Rule is and its importance in calculus. I should mention that it's used for differentiating composite functions, maybe give a simple example.

Next, the detailed sections. The user specified H2 and H3. Let's break down the content. The first H2 could be the mathematical formulation. Here, I need to present the rule mathematically, maybe with both Leibniz and prime notations. Also, explain the steps in applying the rule. Maybe use an example here, like differentiating (3x+2)^5.

Another H2 might be applications. Here, I can talk about why the Chain Rule is important—used in physics, economics, etc. Maybe give a real-world example, like calculating the rate of change in temperature with respect to time when moving through a varying temperature field.

Then, maybe a section on common mistakes or pitfalls. Students often mix up the order of differentiation or forget to multiply by the derivative of the inner function. Highlighting these could be helpful.

Also, an H3 under applications could be higher-order derivatives. Explain how the Chain Rule applies when taking second or third derivatives.

Need to include examples. Maybe a couple of step-by-step problems. Use markdown code blocks for the mathematical expressions, perhaps LaTeX syntax.

For references, list some textbooks or online resources. Maybe include links if possible, but since it's a wiki, perhaps just citations.

I should ensure that technical terms are explained. For instance, when mentioning composite functions, define them briefly. Also, use tables or lists where appropriate, like a list of steps for applying the Chain Rule.

Check the length—needs to be at least 800 characters. Let me estimate. The introduction plus a few sections with examples should cover that. Maybe add more depth in applications or additional examples.

Wait, the user mentioned using code blocks for technical content. Since this is math, LaTeX is probably the way to go. So, equations should be in $$ or using \begin{equation} if needed. But markdown typically uses $ for inline and $$ for block equations. Need to confirm if the markdown supports that.

Also, make sure the structure flows logically. Introduction, then the mathematical formulation, then application steps, examples, common errors, applications in different fields, and maybe higher-order derivatives. Each section with H2 headers, and sub-sections with H3.

Let me start drafting each part. For the introduction, explain that the Chain Rule is a fundamental tool in calculus for differentiating composite functions. Then, in the mathematical formulation, define composite functions, present the rule in both notations, and explain the process. Then examples. Applications can include optimization problems, physics problems. Common mistakes section to warn about typical errors. References at the end.

Make sure the examples are clear. For instance, differentiating sin(x²) or e^(3x). Use multiple examples to cover different cases. Maybe one example with two functions and another with three functions to show the chain rule's extension.

Also, in the pitfalls section, mention differentiating the outer function first and then the inner function, and forgetting to multiply by the inner derivative. Maybe contrast correct and incorrect approaches.

I think that covers the structure. Now, check for markdown syntax. Use # for H1, ## for H2, ### for H3. Lists with hyphens. Code blocks with $$ for equations. Ensure that the LaTeX is correctly formatted. Also, keep paragraphs concise for readability.

Let me start writing each section step by step, ensuring that all requirements are met and the content is accurate and comprehensive.