Page:Calculus Made Easy.pdf/117

Now, before we go on to any further cases, we have two remarks to make. When you are told to equate ${\dfrac {dy}{dx}}$ to zero, you feel at first (that is if you have any wits of your own) a kind of resentment, because you know that ${\dfrac {dy}{dx}}$ has all sorts of different values at different parts of the curve, according to whether it is sloping up or down. So, when you are suddenly told to write

${\frac {dy}{dx}}=0$,

you resent it, and feel inclined to say that it can’t be true. Now you will have to understand the essential difference between “an equation,” and “an equation of condition.” Ordinarily you are dealing with equations that are true in themselves, but, on occasions, of which the present are examples, you have to write down equations that are not necessarily true, but are only true if certain conditions are to be fulfilled; and you write them down in order, by solving them, to find the conditions which make them true. Now we want to find the particular value that $x$ has when the curve is neither sloping up nor sloping down, that is, at the particular place where ${\dfrac {dy}{dx}}=0$. So, writing ${\dfrac {dy}{dx}}=0$ does not mean that it always is $=0$; but you write it down as a condition in order to see how much $x$ will come out if ${\dfrac {dy}{dx}}$ is to be zero.