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 of the arc AP measured from a fixed point A on the curve.
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 (= ).|
From the above figure
Dividing through by , we get
Now let Q approach P as a limiting position; then and we have
Similarly, if we divide (H) by and pass to the limit, we get
Also, from the above figure,
Now as Q approaches P as a limiting position , and we get
- Defined in § 209.