Page:Calculus Made Easy.pdf/30

 depend one on the other. An alteration in one will bring about an alteration in the other, because of this dependence. Let us call one of the variables $$x$$, and the other that depends on it $$y$$.

Suppose we make $$x$$ to vary, that is to say, we either alter it or imagine it to be altered, by adding to it a bit which we call $$dx$$. We are thus causing $$x$$ to become $$x+dx$$. Then, because $$x$$ has been altered, $$y$$ will have altered also, and will have become $$y+dy$$. Here the bit $$dy$$ may be in some cases positive, in others negative; and it won’t (except by a miracle) be the same size as $$dx$$.

Take two examples.

(1) Let $$x$$ and $$y$$ be respectively the base and the height of a right-angled triangle (Fig. 4), of which



the slope of the other side is fixed at $$30^\circ$$. If we suppose this triangle to expand and yet keep its angles the same as at first, then, when the base grows so as to become $$x+dx$$, the height becomes $$y+dy$$. Here, increasing $$x$$ results in an increase of $$y$$. The little triangle, the height of which is $$dy$$, and the base