Rates are merely a way of measuring change. We compare something of interest that is changing compared with something else. Frequently, we are interested in how things change over a period of time (e.g. speed is the distance travelled compared with the time taken for the trip). Sometimes, however, we wish to compare our item of interest with something else, for example we might measure the drop in temperature as we move along a metal rod, away from the fire into which it has been inserted (this would be a change in temperature with distance).
When you graph a straight line on graph paper, its gradient is a rate. In calculus, the derivative dy/dx is a rate. Rates are everywhere around us, from pay rates to electricity and water consumption to data transfer rates on your electronic devices, to how much you pay for your phone plan (e.g. dollars per month).
Rates always compare things that are of a different kind (comparing like things would be considered a ratio by mathematicians). Rates are written as fractions and we can therefore manipulate them in exactly the same way that we manipulate fractions. Because they are written as fractions, when we read them aloud, we replace the fraction bar (the vinculum) with the word per. If you hear (or see) the Latin word per appear in conversation (or on paper) that is almost a guarantee that you are dealing with a rate … dollars per hour, degrees per metre, kilometres per hour, tonnes per cubic metre, revolutions per second, etc.
By the way, if you drive a car, you must be familar with rates. Your speed is a rate; your fuel consumption is a rate; and your tachometer measures a rate. The image I chose for this page is a screen grab from a YouTube video in which Ed Reiner demonstrates a car heads-up-display of his own design. It seemed very appropriate for this topic.
Sir frankly, I only wanted to learn the chain rule first but your videos are so good and so amazing that I feel like studying calculus more and more but only nd only from your videos. can you do me a favor and make a video for log functions differentiation please? it would be a great help 🙂
Sudhanshu K (on a CCM video about Implicit Differentiation Made Simple)
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