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628 MOLESWORTH S POCKET-BOOK
Differential and Integral Calculus—continued.
Fobmulæ for Successive Differentiation.
Let N = number of times to which successive differentiation is tv
be carried.________________________________________________
If u =
Xn
a*
sin. nr
log. x
d™ u
then -— =
d æN
’ n(n-l).........[n-(N-l)J»»-N
(- i)n n(n + 1)....n + (N — !)»-(«+»)
(log.e a)« ax
nN sin. (na; + f Nir)
(_ i)n -1 (N - 1) (N - 2)_______3.2.1.«-"
2(1 -»)-(* + !) N(N-l) (N -2)____3.2.1
Taylob’s and Maclaurin’s Theorems.
Let y be a function of x which it is possible to develop in a series of
ascending powers of that variable; and suppose that h — any indeter-
minate quantity, y = A + Bæ + Qx* + D®3 + E®4 + &c- i and when
x becomes x + h, let y —
dy , d^y
y y dx dx2
3/',
h2 d3y h3 + .
1X2+ <fc3 1X2X3 da*
or “ Taylor’s Theorem ” ;
. _L_Æ) .x* + —— ■
+ lX2\dæ2'o 1X2X3
or “ Maclaurin’s Theorem.”
hi .
------ — -f~ &C.j
1X2X3X4
. /dx\
2', = ^+U)o
Æ)
\dx*J
___________
■x.dx _ log, Xi _ hyp log. x _ hyp log.
x„ x
Integration by I’ahts.
Simpson’s Fobmula of Quadratures.
To find the approximate value of any integral of
the form/z.dx, where z is any function of x.
Find n values of z, corresporiding to n equidistant values of x; such
as zn z,z.,z,.....Zn-„ Zn-1 zn; then the value of the Integral
is the product of the third part of the equidistance, by a sum com-
pounded of, (A) the extreme values of z; or (z0 + «„); (B) the quad
rupie sum of the odd values of 2; or i (zt + z3 +.«»-ij i (V)
the double sum of the even values of z; or 2 (z2 + zt + ... ■ Zn _ J;
the greater the value of n, the nearer will be the approximation.