Chapter 33
Differentiation of inverse
trigonometric and
hyperbolic functions
33.1 Inverse functions
If y = 3x − 2, then by transposition, x =
y + 2
3
. The
function x =
y + 2
3
is called the inverse function of
y = 3x − 2 (see page 188).
Inverse trigonometric functions are denoted by prefixing the function with ‘arc’ or, more commonly, by
using the −1 notation. For example, if y = sin x, then
x = arcsin y or x = sin −1 y. Similarly, if y = cos x, then
x = arccos y or x = cos −1 y, and so on. In this chapter
the
−1 notation will be used. A sketch of each of the
inverse trigonometric functions is shown in Fig. 33.1.
Inverse hyperbolic functions are denoted by prefixing the function with ‘ar’ or, more commonly, by
using the −1 notation. For example, if y = sinh x, then
x = arsinh y or x = sinh
−1 y. Similarly, if y = sech x,
then x = arsech y or x = sech −1 y, and so on. In this chapter the −1 notation will be used. A sketch of each of the
inverse hyperbolic functions is shown in Fig. 33.2.
33.2 Differentiation of inverse
trigonometric functions
(i) If y = sin
−1 x, then x = sin y.
Differentiating both sides with respect to y gives:
dx
dy
= cos y =
1 − sin 2 y
since cos 2 y + sin 2 y = 1, i.e.
dx
dy
=
√
1 − x 2
However
dy
dx
=
1
dx
dy
Hence, when y = sin
−1 x then
dy
dx
=
1
√
1 −x 2
(ii) A sketch of part of the curve of y = sin
−1 x is
shown in Fig. 33.1(a). The principal value of
sin −1 x is defined as the value lying between
−π/2 and π/2. The gradient of the curve between
points A and B is positive for all values of x
and thus only the positive value is taken when
evaluating
1
√
1 − x 2
(iii) Given y = sin −1 x
a
then
x
a
= sin y and
x = a sin y
Hence
dx
dy
= a cos y = a
1 − sin 2 y
= a
1 −
x
a
2
= a
a 2 − x 2
a 2
=
a
√
a 2 − x 2
a
=
√
a 2 − x 2