I assigned another very easy but good problem this week. It was simple enough, but it gave a hint of things to come.

Use the Product Rule to find the derivative of .

Since we have not yet discussed the Chain Rule, the Product Rule was the only way to go.

And likewise for higher powers:

If you just look at the answer, it is not clear where the comes from. But the result foreshadows the Chain Rule.

Then we used the new formula to differentiate a few expressions such as and and a few others.

Regarding the Chain Rule: I have always been a proponent of the Rule of Four, but I have never seen a good graphical explanation of the Chain Rule. (If someone has one, PLEASE send it to me – I’ll share it.)

Here is a rough verbal explanation that might help a little.

Consider the graph of . On the interval it goes through all its value in order once – from 0 to 1 to 0 to -1 and back to zero. Now consider the graph of . On the interval it goes through all the same values in one-third of the time. Therefore, it must go through them three times as fast. So the rate of change of between 0 and must be three times the rate of change of . So the rate of change of must be . Of course this rate of change is the slope and the derivative.

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