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In computer programming there's an idea called "top-down design" or "stepwise refinement." This is essentially the art of breaking a big problem down into little problems, then breaking the little problems into itty-bitty problems, and continuing to break the problems down until they're small enough to deal with.
Example. This page has a good example of top-down design. The big problem is to draw a house. This can be broken into the littler problems of drawing the outline, the door, and the windows. The problem of drawing the windows can be broken down into one problem for each window, and so on.
The problem of finding the derivative of a product fg can be broken down into several smaller problems:
We're not only practicing taking derivatives here. We're practicing the skill of breaking problems down into smaller problems, and this skill will be valuable everywhere else in your life.
Think about the big problem of "doing your homework." This can be broken down into the smaller problems of math homework, that English paper, studying for the Spanish test, etc.. That English paper can be broken down into the even smaller problems of writing an introduction, writing 3 middle paragraphs, and writing a conclusion. Suddenly the problems are much more manageable - instead of trying to sit down and "do your homework," you can sit down and write the first middle paragraph. Then you can cross something off your to-do list and feel good about yourself!
When Leibniz created his dy/dx notation, he did mean for dy and dx to be numbers - infinitesimal numbers, that is. An infinitesimal number is a number that's bigger than zero but smaller than every positive real number. Think about that for a minute. If ε is infinitesimal, then ε is greater than zero but smaller than 0.00001, smaller than 0.000001, smaller than 0.00000000000001, etc. No matter how many zeros there are, ε < 0.00...001
Mathematicians had some trouble with the idea of infinitesimals, because it seemed too imprecise and fuzzy. They used limits instead, and we still define continuity and derivatives in terms of limits. It wasn't until 1966 that a guy named Robinson wrote a book called "Non-standard Analysis" and convinced people that yes, we could do calculus properly with these weird infinitesimal numbers.
There are three steps to solving a math problem.
Find the derivative of the function h(x) = cos(sin(ln x))).
We need to use the chain rule again to find (sin (ln x))'. The outside function is now sin(□) and the inside function is ln x, therefore
Now we can go back to our first application of the chain rule and simplify: