Page:Calculus Made Easy.pdf/33

 a mere ratio, namely, the proportion which $$dy$$ bears to $$dx$$ when both of them are indefinitely small.

It should be noted here that we can only find this ratio $$\frac{dy}{dx}$$ when $$y$$ and $$x$$ are related to each other in some way, so that whenever $$x$$ varies $$y$$ does vary also. For instance, in the first example just taken, if the base $$x$$ of the triangle be made longer, the height $$y$$ of the triangle becomes greater also, and in the second example, if the distance $$x$$ of the foot of the ladder from the wall be made to increase, the height $$y$$ reached by the ladder decreases in a corresponding manner, slowly at first, but more and more rapidly as $$x$$ becomes greater. In these cases the relation between $$x$$ and $$y$$ is perfectly definite, it can be expressed mathematically, being $$\frac{y}{x}=\tan 30^\circ$$ and $$x^2+y^2=l^2$$ (where $$l$$ is the length of the ladder) respectively, and $$\frac{dy}{dx}$$ has the meaning we found in each case.

If, while $$x$$ is, as before, the distance of the foot of the ladder from the wall, $$y$$ is, instead of the height reached, the horizontal length of the wall, or the number of bricks in it, or the number of years since it was built, any change in $$x$$ would naturally cause no change whatever in $$y$$; in this case $$\frac{dy}{dx}$$ has no meaning whatever, and it is not possible to find