The Idea of the Derivative

In Chapter 4 we estimated that in 2017 $$CO_2$$ C O 2 levels were increasing at a rate of 2.32 ppm per year. This is an instantaneous measure of change or in a sense the speed of the function at 2017. At this point the 2.32 ppm is an estimate of the slope of the tangent line. Before we turn to calculating tangent line slopes explicitly we want to introduce terminology and notation for the slope of the tangent line and then develop an understanding of the information provided by the slope of the tangent line. In M-Box 5.1 we introduced notation for the derivative or the instantaneous rate of change at a point. In the context of CO2 we write $$CO2'(67) \approx 2.32$$ C O 2 ′ ( 67 ) ≈ 2.32 ppm per year. The prime notation is used to represent the instantaneous rate of change of a specific original function, in this case CO2(t). Note the relationship between the function and the derivative of the function. We use f(a) for the value of the function at $$x=a$$ x = a and $$f'(a)$$ f ′ ( a ) for the derivative or the instantaneous rate of change of the function f(x) at $$x=a$$ x = a .