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.

~ 29-31.

Trigonometric and inverse functions.

55

y) y = tan -1 x. Inverse function: x = tan y.
dy
d - (cos )2 -1.
6) y = cot- x. Inverse function x = coty.
dy i
d- = -(sin i) - 1+ x2
In the first two of these functions the values + 1, at which the
definition of the functions ceases, form special points; in the last two,
x goes from - o to -+ o and the functions as well as their
differential quotients are finite even at these limits.
The logarithm and the circular functions are transcendental; but
their differential quotients are algebraic.
31. But lastly we can also differentiate the explicit irrational
function: y = —xm where m means any real number, but x is
positive, and the root is always taken positively. For, taking the
natural logarithm of both sides of the equation y = xm we have
1(y) =ml(x). If we differentiate this equation, remembering that
y on the left side is a function of x, it follows by Rule 5) ~ 26 that
1 d m- therefore: d-y ==- m =
y dx x dx x
We have accordingly for every value of m the equation:
dy- d(xm) = m rn-l, (y > 0).
dx dx
It is to be noticed, that when 0 < m < 1 the function is finite for
x - 0, and infinitely great for x=- ~+ o, whereas its differential
quotient is infinitely great at the former point and at the latter finite
and equal to zero. For the Implicit Algebraic Function see Chap. X.
For all functions dealt with in this Chapter it is indifferent
whether Ax is chosen positive or negative; i. e. all these functions
have at each point the value of the progressive differential quotient
equal to that of the regressive; each has a derived function or derivate.