Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/158

90. Derivative of the arc in rectangular coördinates. Let s be the length^[1] of the arc AP measured from a fixed point A on the curve.

Denote the increment of s (= arc PQ) by $\Delta s$. The definition of the length of arc depends on the assumption that, as Q approaches P,

$\lim \left({\frac {{\mbox{chord}}PQ}{{\mbox{arc}}PQ}}\right)=1.$

If we now apply the theorem in §89 to this, we get

(G)	In the limit of the ratio of chord PQ and a second infinitesimal, chord PQ may be replaced by arc PQ (= $\Delta s$).

From the above figure

(H)	$({\mbox{chord}}PQ)^{2}=(\Delta x)^{2}+(\Delta y)^{2},$

Dividing through by $(\Delta x)^{2}$, we get

(I)	$\left({\frac {{\mbox{chord}}PQ}{\Delta x}}\right)^{2}=1+\left({\frac {\Delta y}{\Delta x}}\right)^{2}$.

Now let Q approach P as a limiting position; then $\Delta x{\dot {=}}0$ and we have

$\left({\frac {ds}{dx}}\right)^{2}=1+\left({\frac {dy}{dx}}\right)^{2}$.

[Since $\lim _{\Delta x\to 0}\left({\tfrac {chordPQ}{\Delta x}}\right)=\lim _{\Delta x\to 0}\left({\tfrac {\Delta s}{\Delta x}}\right)={\tfrac {ds}{dx}}$, (G).]

(24)	∴ ${\frac {ds}{dx}}={\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}.$

Similarly, if we divide (H) by $(\Delta y)^{2}$ and pass to the limit, we get

(25)	${\frac {ds}{dy}}={\sqrt {\left({\frac {dx}{dy}}\right)^{2}+1}}.$

Also, from the above figure,

$\cos \theta ={\frac {\Delta x}{{\mbox{chord}}PQ}}$ $\sin \theta ={\frac {\Delta y}{{\mbox{chord}}PQ}}.$

Now as Q approaches P as a limiting position $\theta {\dot {=}}\tau$, and we get

(26)	$\cos \tau ={\frac {dx}{ds}}$, $\sin \tau ={\frac {dy}{ds}}.$

[Since from (G) $\lim {\tfrac {\Delta x}{{\mbox{chord}}PQ}}=\lim {\tfrac {\Delta x}{\Delta x}}={\tfrac {dx}{ds}}$, and $\lim {\tfrac {\Delta y}{{\mbox{chord}}PQ}}=\lim {\tfrac {\Delta y}{\Delta s}}={\tfrac {dy}{ds}}$.]

1. Defined in § 209.