The next evening Sarah was washing dishes while her mother worked at the table. Of all the household chores, Sarah disliked this one the least. Particularly in the summer, when the windows were open and you could hear snatches of conversation, radio and TV sounds, and, if you were lucky, an occasional bird calling.
"Well, Sarah, I have read all that stuff in the Reader; I guess I know what an instantaneous rate of change is. And I like to think about the derivative as the slope of the line when you zoom in on the graph. I suppose I can deal with the different notations, though I am not quite sure what a differential is. But I only know how to calculate the derivative of linear functions and things like kt2. You know, I've been working with Anthony on his precalculus ... "
Anthony is Sarah's brother. He's into basketball, rap music, and impressing girls; precalculus is not high on his list of priorities. Sarah's mother had been working with him almost every night — in Sarah's view, with amazing patience. Sarah had heard them when she was home for spring break. Her mother would try to explain the concept behind the particular problem; Anthony would break in, saying that he just wanted to write down solutions and quit.
Sarah's mother continued, "How do you calculate the derivatives of some of the other functions you see in precalculus — say, a third-degree polynomial like 5t3 + 7t2 - 3t + 6?"
Sarah replied, "That's not too hard. You need to learn the rule for differentiating powers and then just use the rules for derivatives of sums and constant multiples."