Page:Calculus Made Easy.pdf/46



our equations we have regarded $$x$$ as growing, and as a result of $$x$$ being made to grow $$y$$ also changed its value and grew. We usually think of $$x$$ as a quantity that we can vary; and, regarding the variation of $$x$$ as a sort of cause, we consider the resulting variation of $$y$$ as an effect. In other words, we regard the value of $$y$$ as depending on that of $$x$$. Both $$x$$ and $$y$$ are variables, but $$x$$ is the one that we operate upon, and $$y$$ is the “dependent variable.” In all the preceding chapter we have been trying to find out rules for the proportion which the dependent variation in $$y$$ bears to the variation independently made in $$x$$.

Our next step is to find out what effect on the process of differentiating is caused by the presence of constants, that is, of numbers which don’t change when $$x$$ or $$y$$ change their values.

Added Constants.

Let us begin with some simple case of an added constant, thus: Let $y=x^3+5$. Just as before, let us suppose $$x$$ to grow to $$x+dx$$ and $$y$$ to grow to $$y+dy$$.