What are some techniques that can be employed to best develop sentences and convey thoughts accurately and in an organized manner?
Truthfully, the answer to this question depends on a few different components -- the subject, the motivation and purpose for writing, the specific style requested, and the writer themselves. Some writers will find that they are more productive if they just begin writing and get all of their ideas on the page to be edited, rearranged, and modified. Others may find it easier to begin with an outline of ideas that contains the main concepts, the order in which they are introduced, and supporting ideas. Further, moving beyond these two possibilities, the subject at hand may require a specific style of writing (e.g. the style for the American Chemical Society) and may have different requirements for conveying a thought completely, such as with a well formed introduction and explanation or equation or with a quotation and corresponding analysis.
Why is it important to understand the process for establishing a derivative rather than starting by memorizing common derivative forms or different shortcuts?
A huge part of calculus is understanding the relationship between variables, between functions, and between rates. Instances may arise where there is no familiar relationship or shortcut for a problem, and the individual is left to determine a model or relations with the information given. With this task at hand, I would assert that the process of learning how to take a derivative is an important step in developing the mindset and problem solving skills that may be needed later to relate different functions and variables. Even moreso, derivatives are arguably the entire basis for calculus, and not fully understanding what they mean and where they come from will prevent an individual from truly grasping more difficult concepts within the field.
How does an individual's word choice affect the subconscious understanding and feelings of the intended audience?
While the question presents an opportunity to discuss the connotation of words in spite of their simple definition, I believe the easiest way to get the point across is with an example. On the most basic level, this can be the difference in recognizing how the choice between using the articles "a" or "an" and using the article "the." The average person wouldn't think twice before picking which article to use when introducing a noun, but these words hold importance that employ without realizing it. For example, in listing an item of my possessions, I may include a water bottle, a phone, a painting, and a wallet. Alternatively, I may list these items as: a water bottle, a phone, the painting, and a wallet. The lists, though almost identical, differ in one instance. "A painting" presents the idea that the painting may be generally unimportant to me, may be one of my many paintings, or may not hold relevance to the surrounding context. The choice of using "the painting," however, brings importance to the word. Why "the?" Is it my only painting? Is it my favorite painting? Is it the painting I mentioned earlier? Or is it a painting that I am about to explain the importance of?