On Relative Growings

All through the calculus we are dealing with quantities
that are growing, and with rates of growth.
We classify all quantities into two classes: *constants*
and *variables*. Those which we regard as of fixed
value, and call *constants*, we generally denote algebraically
by letters from the beginning of the
alphabet, such as $a$, $b$, or $c$; while those which we
consider as capable of growing, or (as mathematicians
say) of “varying,” we denote by letters from the end
of the alphabet, such as $x$, $y$, $z$, $u$, $v$, $w$, or sometimes $t$.

Moreover, we are usually dealing with more than one variable at once, and thinking of the way in which one variable depends on the other: for instance, we think of the way in which the height reached by a projectile depends on the time of attaining that height. Or we are asked to consider a rectangle of given area, and to enquire how any increase in the length of it will compel a corresponding decrease in the breadth of it. Or we think of the way in which any variation in the slope of a ladder will cause the height that it reaches, to vary.

Suppose we have got two such variables that
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 $dx$.

(1) Let $x$ and $y$ be respectively the base and the height of a right-angled triangle (Figure 4), of which the slope of the other side is fixed at $30°$. 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 of which is $dx$, is similar to the original triangle; and it is obvious that the value of the ratio $\dfrac{dy}{dx}$ is the same as that of the ratio $\dfrac{y}{x}$. As the angle is $30°$ it will be seen that here \[ \frac{dy}{dx} = \frac{1}{1.73}. \]

(2) Let $x$ represent, in Figure 5, the horizontal distance, from a wall, of the bottom end of a ladder, $AB$, of fixed length; and let $y$ be the height it reaches up the wall. Now $y$ clearly depends on $x$. It is easy to see that, if we pull the bottom end $A$ a bit further from the wall, the top end $B$ will come down a little lower. Let us state this in scientific language. If we increase $x$ to $x + dx$, then $y$ will become $y - dy$; that is, when $x$ receives a positive increment, the increment which results to $y$ is negative.

And the ratio of $dy$ to $dx$ may be stated thus: \[ \frac{dy}{dx} = - \frac{0.11}{1}. \]

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 $\dfrac{dy}{dx}$ has
no meaning whatever, and it is not possible to find
an expression for it. Whenever we use differentials
$dx$, $dy$, $dz$, etc., the existence of some kind of
relation between $x$, $y$, $z$, etc., is implied, and this
relation is called a “function” in $x$, $y$, $z$, etc.; the
two expressions given above, for instance, namely
$\dfrac{y}{x} = \tan 30°$ and $x^2 + y^2 = l^2$, are functions of $x$ and $y$.
Such expressions contain implicitly (that is, contain
without distinctly showing it) the means of expressing
either $x$ in terms of $y$ or $y$ in terms of $x$, and for
this reason they are called *implicit functions* in
$x$ and $y$; they can be respectively put into the forms
\begin{align*}
y &= x \tan 30° \quad\text{or}\quad x = \frac{y}{\tan 30°} \\
\text{and}\;
y &= \sqrt{ l^2 - x^2} \quad\text{or}\quad x = \sqrt{ l^2 - y^2}.
\end{align*}

These last expressions state explicitly (that is, distinctly)
the value of $x$ in terms of $y$, or of $y$ in terms
of $x$, and they are for this reason called *explicit
functions* of $x$ or $y$. For example $x^2 + 3 = 2y - 7$ is
an implicit function in $x$ and $y$; it may be written
$y = \dfrac{x^2 + 10}{2}$ (explicit function of $x$) or $x = \sqrt{2y - 10}$
(explicit function of $y$). We see that an explicit
function in $x$, $y$, $z$, etc., is simply something the
value of which changes when $x$, $y$, $z$, etc., are
changing, either one at the time or several together.
Because of this, the value of the explicit function is
called the *dependent variable*, as it depends on the
value of the other variable quantities in the function;
these other variables are called the *independent
variables* because their value is not determined from
the value assumed by the function. For example,
if $u = x^2 \sin \theta$, $x$ and $\theta$ are the independent variables,
and $u$ is the dependent variable.

Sometimes the exact relation between several quantities $x$, $y$, $z$ either is not known or it is not convenient to state it; it is only known, or convenient to state, that there is some sort of relation between these variables, so that one cannot alter either $x$ or $y$ or $z$ singly without affecting the other quantities; the existence of a function in $x$, $y$, $z$ is then indicated by the notation $F(x, y, z)$ (implicit function) or by $x = F(y, z)$, $y = F(x, z)$ or $z = F(x, y)$ (explicit function). Sometimes the letter $f$ or $\phi$ is used instead of $F$, so that $y = F(x)$, $y = f(x)$ and $y = \phi(x)$ all mean the same thing, namely, that the value of $y$ depends on the value of $x$ in some way which is not stated.

Let us now learn how to go in quest of $\dfrac{dy}{dx}$.

It will never do to fall into the schoolboy error of thinking that $dx$ means $d$ times $x$, for $d$ is not a factor–it means “an element of” or “a bit of” whatever follows. One reads $dx$ thus: “dee-eks.”

In case the reader has no one to guide him in such
matters it may here be simply said that one reads
differential coefficients in the following way. The
differential coefficient
$\dfrac{dy}{dx}$
is read “*dee-wy by dee-eks,*” or “*dee-wy over dee-eks.*”

So also
$\dfrac{du}{dt}$ is read “*dee-you by dee-tee.*”

Second differential coefficients will be met with
later on. They are like this: $\dfrac{d^2 y}{dx^2};$
*which is read “dee-two-wy over dee-eks-squared,”*
and it means that the operation of differentiating $y$
with respect to $x$ has been (or has to be) performed
twice over.

Another way of indicating that a function has been
differentiated is by putting an accent to the symbol of
the function. Thus if $y=F(x)$, which means that $y$
is some unspecified function of $x$ (see here), we may
write $F'(x)$ instead of $\dfrac{d\bigl(F(x)\bigr)}{dx}$. Similarly, $F”(x)$
will mean that the original function $F(x)$ has been
differentiated twice over with respect to $x$.

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