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Study Guide

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:

- find
*f*'

- find
*g*'

- apply the product rule

- simplify

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!

### I Like Abstract Stuff; Why Should I Care?

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...001Mathematicians 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.

### How to Solve a Math Problem

There are three steps to solving a math problem.

- Figure out what the problem is asking.

- Solve the problem.

- Check the answer.

### Sample Problem

Find the derivative of the function

*h*(*x*) = cos(sin(*ln**x*))).Answer.

- Figure out what the problem is asking.

The problem is asking us to find a derivative. More specifically, it's asking us to use the chain rule to find a derivative. We can tell this because there are nested functions. In fact, the functions are nested 3 deep. We will use the chain rule 2 times to find the derivative we want.

- Solve the problem.

cos(□) is the outside function and {sin(*ln**x*)} is the inside function. Then the chain rule says

*h'*(*x*) = -sin({sin(*ln**x*)}) × ({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, thereforeNow we can go back to our first application of the chain rule and simplify:

- Check the answer.

There's no great way to check this answer. We could do it over again and make sure we find the same answer the second time; we could check the answer in the book; we could have a calculator or computer compute the derivative and compare it to our answer.

- Figure out what the problem is asking.

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