An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

26

SUMMATION OF DIFFERENCES

Another notation frequently employed is explained by either
of the definitions
d Lt _ _ _ _ _ _ _ _ _
f'(0)- x/(X.), '()=X) Lt ( /)The slope of the curve y = (x) may be expressed by any
one of
d dy Lt Xtr(x +/) — ' (x) Lt Ay
dx (Y) = -' -o ---- ---- Aro a', (-'.
EXERCISES
1. (x 3+4)33( )2+; d e=ex.
dx x 3;
dax
2. y-=snax. — = a COS x.
3. f(x) = x'", n being a positive integer. f' (x) = nx"-1.
4. - cos ax - - sin ax; - tan x = sec2 x.
dx dx
X + I 2
5. (x) =. (X)= (_ )2
x- I
6. y=log,az. t'(Z)= T
z
d d _
7. - - =? - 11Z-?
dz dmi
8. If y = log 4 (x), prove dy = '(x.
dx (x)
SOLUTION.
Ay I o b (x + /)
-ILog (+ + ~(X+ h)-+ (x) h1
=- )log [(I + Z)1]'()+ )log [(I + Z)Z] Z]hO
(x),.ty=q'(x)
dx ~(x)