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.

54

Differentiation.

1Bk. I ch VI.

4 sin Je Ax sin (x I cAx)
Afor ) y cos (x + A-x) - cos sin A Sil (X + x)
for p)- Ax
dLi =( - nx) Lim) L sin (x + -1 Ax) -sin x
dx - ~ Ax
d sin x d cos x
cos x -- -sin
sin x dy dx cdx?) y\ f-tangx -, --
y) = tan x coS x dx (cos X)2
dy 1
dx (cosx)2
d cos x d sin x
sin x -- cos x CO s
cos x d y dx dx
6) y -- cot x —
sin x dx (sin X)2
dy 1
dx (sin )2
All these four as well as their derived functions are quite indeterminate for infinite values of x. For all arguments for which the last
two functions are infinite their derived functions are also infinite.
30. Inverse functions: the logarithm and the circular
functions.
dx
General rule: If x f(y), and dy =f'(y) is calculated, then
if y = (x) express the inverse function (~ 13), it follows that
dy (x) = _ For we have: d c Lim Y =- Lim (: Ax
dax f' (y) dx Ax Ay
We calculate therefore p'(x) for any value of x, by substituting in the
expression,r ) the corresponding value of y; points at which f'(y) = 0,
demand special attention.
1. y == log x. Inverse function: x = a. (a > 0).
dy _ 1_ 1 t log
ix -Y ~ icc =2: ~L, 7 log e.
dx (ay I a x la x '
2. a) y -= sin- x. Inverse function: x = sin y.
dy I _
dx cos y y _ -
The square root is positive because by our convention (~ 13)
- - < y <~ 7- therefore cos y > 0.
f) y = cos-Ix. Inverse function: x = cos y.
dy 1
d x sin y _-. '
The square root is positive because
0 < y < rf therefore sin y > 0.