Section Summary

In this section we have given a name — differential — and a notation — `dy` — to an idea we have seen frequently, that of linear approximation to change in nonlinear functions. We also have seen that differentials can be used for quick estimates of changes. Suppose the dependent variable `y` is a function of the independent variable `t`, say,

The differential `dt` represents a (possibly small) change in `t` from a particular value, say, `t_1.` The corresponding differential `dy` is the corresponding change in `y` along the line tangent to the graph of `y=ftext[(]t text[)]` at $(t_1,f\left(t_1\right))\mathrm{.}$ Since the slope of this line is $f^{\prime }\left(t_1\right)$, we have

Thus we get a quick estimate of change in `y` as slope `times` run.