An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 35. 36.

Recurring and independent formulas.

63

The calculation of the nth derived function y(f) from the last equation
requires that of all preceding derivates from the preceding equations;
the formula established for y(n) is called on this account a recurring
formula. All the derived functions are finite, except where cos x = 0.
5) y =cot x, y sin x = os x.
y(s)sinx -+ tn y(n-1) sin (X + rX) + 2(2)sin( + 2- ) +
+ nky(n- ) sin(x -+ k )... -f- ysin (x +- n r) = cos(x 4- - nz).
y Zg 2 d _3y _.2
III. I) y "log, dogy x "l, -og ge - og e,
dx X dX'dx dx d x
dy - ( 1)("-i) alog e. (a > 0.)
d xn x
2) y == sin-lx, (- 1 < x < - 1, 2< < + i-)
From the equation: y'= -1 i.e. y'/1 - x-= 1, follows on further
/e - x2
differentiation, the quantity on the right being constant:
y"J/1 -x - _- =0 or y"(1 -x2) — y'x = 0.
Y1/ - x~
Differentiating this equation n times by the Rule of the Product, we find
y(n+2)(l - x2) = (2n + 1)y(n+l)x + n2y(yn)
This is likewise a recurring formula for the calculation of all the
derived functions; they become infinite for the arguments x2= 1.
dny dn sin —1 x
3) y = cos-Ix = - sin-'x. dy d
d n X' ~dxn
4) y = tan —'x, (- 2 < y < + ').
From the equation dy,, or y'( + x2), follows:
y(n+l) (1 + X2) + 2n1ly(n)X' - 2n2y-(n — ) = 0 or:
y(n+l) (1 - X2) -- 2 n x y() - n (n - 1) y(-); and lastly
K\) jor JL 1 cot-x - 1 nY d'tan1 X
5) for y- = cot-x =- -- tan-1x, we have =- -ta- - ~
dxn d x"
36. For circular functions we have thus found only recurring
formulas; such formulas we shall obtain for every compound function
by applying the Rule of the Product. But we can also propose the
problem: to calculate the nth derivate by an independent formula
not requiring first the calculation of all preceding derived functions.*)
As an example of obtaining an independent expression, we can
treat y -- tan-~x in the following particularly simple manner.
*) The propositions bearing on this are discussed in detail in SchloSilch:
Compendium der hoheren Analysis, Vol. II, and Hoppe: Theorie der hoheren
Differentialquotienten.