Page:Calculus Made Easy.pdf/39

 What does $$(dx)^2$$ mean? Remember that $$dx$$ meant a bit–a little bit–of $$x$$. Then $$(dx)^2$$ will mean a little bit of a little bit of $$x$$; that is, as explained above (p. 4), it is a small quantity of the second order of smallness. It may therefore be discarded as quite inconsiderable in comparison with the other terms. Leaving it out, we then have: $y+dy=x^2+2x \cdot dx$. Now $$y=x^2$$; so let us subtract this from the equation and we have left $dy=2x \cdot dx$. Dividing across by $$dx$$, we find $\frac{dy}{dx}=2x$. Now this is what we set out to find. The ratio of the growing of $$y$$ to the growing of $$x$$ is, in the case before us, found to be $$2x$$.