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