192 
DERIVATIVES AND INTEGRALS 
[VI 
two valued on one side of P and not defined on the other. In 
such cases we may consider the two sets of values of <j>{x), which 
occur on one side of P or the other, as defining distinct functions 
(jrfx) and </> 2 («), the upper part of the curve corresponding to (fjfx). 
The reader will easily convince himself that in (a) 
{<p(x + h) — <f)(x)]/h-* + oo , 
as h-*~ 0, and in (b) 
{<]>(x + h) — (p(x)]/h — oo ; 
while in (c) 
{(f>i(x + h) - (pfx)]/h-*- + go , + h) — (p 2 {x)]/h — oo , 
and in (d) 
{(pfx + h) —(^(x^/h-*- — x , [<p. 2 (x + h) — <j> 2 (x)]/h-*- + oo, 
though of course in (c) only positive and in (d) only negative 
values of h can be considered, a fact which by itself would preclude 
the existence of a derivative at P. 
94. Some general rules for differentiation. Through 
out the theorems which follow we assume that the functions 
f(x) and F (x) have derivatives f\x) and F'(x), for the values of 
x considered. 
(1) If <f>(x) =/{x) + F (x), (f){x) has a derivative 
cf>'{x)=f\x) + F'{x). 
(2) If (f){x) = cf{x), where c is a constant, (f>{x) has a de 
rivative <t>\v) = cf\x). 
We leave it as an exercise to the reader to deduce these results 
from the general theorems on limits stated in Ex. xxxvi. 1. 
(3) If (f) (x) = f(x) F{x), (f) (x) has a derivative 
*'(«)-/(*) F\x)+f'{x) F(x). 
For <f)' (x) = lim [f{x + h) F (x + h) — f{x) F (x)]/h 
= Urn i fix + h) F ^ + h)-F(x) + F{x) fix + h) -fix) | 
=/0) F'{x) + F{x)f{x). 
(4) <f) (x) — 1 ff(x) has a derivative f {x) = — f'{%)/{f(x)] 2 .