related rates

Everyone's favorite genre of calculus problems. A sample problem might look like this:

Joe is flying his airplane 250 miles above the ground at a speed of 400 miles an hour towards point A on the ground. At the instant Joe is 50 miles away from point A, how fast is the angle of decent between the plane and point A changing?

A type of calculus problem which is an application of the derivative. The problems themselves are not hard, but the setup of the problem can often cause many a college student (or high school student) to rip their textbook in half in a rage of frustration.

The idea of a related rates problem is to compute the change in one quantity in terms of the rate of change in another quantity. The procedure is as follows:

1) Find an equation that relates the two quantities in question to each other
2) Use the Chain Rule to differentiate both sides of the equation with respect to time (time just gets thrown in the equation, as everything happens in our stream of time).

Problems are usually along the lines of a balloon filling up, a shadow and a man running from a light post, or a rope pulling in a boat from a harbor. The problems apply simple concepts, and can add up to a challenging time, but if done diligently, can really add to one's understanding of calculus.

These are some systematic, clear steps that students can take to help in solving a related rates problem:

Step 1, Draw a figure and label the quantities that vary.
In a related rates problem, you'll always have at least two varying rates. Sometimes more. Keeping track of these is difficult, but vital to solving the problem, so don't muck things up from the beginning by trying to figure it out in your head. If a figure is provided, redraw the figure and label what information was provided in the problem's setup. If not, imagine the scene described and draw the best you can. It will at least get you in the right frame of mind to have to consider how things are placed, even if the path seems clear. You may catch something you might've missed. Besides all that, it looks purtty on paper.

Step 2, Identify the rates of change that are known and the rate of change that is to be found.
It sucks to grind through a problem missing a piece of information that you could've known from the beginning. Don't let it happen, keep your bases covered by writing out everything you know for sure. This is important in any word problem, but especially for related rates, because you will not get anywhere if you miss a provided piece of information. They're designed that way (the bastards!). Be careful, though, of extraneous info.

Step 3, Find an equation that relates the quantity whose rate of change is to be found to the quantities whose rates of change are known.
This will be a set-up with static variables, like just plain ol' x, y, or z. One variable may also be something more complicated, like volume, surface area, or angle. Just make sure it's accurate. You need this equation for the next step.

Step 4, Differentiate both sides of this equation with respect to time and solve for the derivative that will give the unknown rate of change.
You're almost homefree, substitute substitue substitute! This is when you take that nice list of known things at the beginning and plug it in like mad. Hopefully (prey hard), you'll just have your unknown rate of change left. It can either be a constant, or an equation in terms of the variable defining the rate of change. If the latter, there's one last step.

Step 5 (optional), Evaluate this derivative at the appropriate point.
Plug, chug, scratch down the answer and you're done. Congrats! Time to move on to the next one... *sigh*

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Steps sourced from Anton, Howard. Calculus: A New Horizon, 6th ed. New York: John Wiley & Sons, 1999. Explication my own work