Page:Calculus Made Easy.pdf/34

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 ${\frac {y}{x}}=\tan 30^{\circ }$ 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

$y=x\tan 30^{\circ }$ or $x={\frac {y}{\tan 30^{\circ }}}$

and

$y={\sqrt {l^{2}-x^{2}}}$ or $x={\sqrt {l^{2}-y^{2}}}$.

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 $y={\frac {x^{2}+10}{2}}$ (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;