Chapter 33
Differentiation of inverse
trigonometric and
hyperbolic functions
33.1 Inverse functions
Ify= 3 x−2, then by transposition,x=
y+ 2
3
.The
functionx=
y+ 2
3
is called theinverse functionof
y= 3 x−2 (see page 188).
Inverse trigonometric functionsare denoted bypre-
fixing the function with ‘arc’ or, more commonly, by
using the−^1 notation. For example, ify=sinx,then
x=arcsin y orx=sin−^1 y. Similarly, ify=cosx,then
x=arccosyorx=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 functionsare denoted by pre-
fixing the function with ‘ar’ or, more commonly, by
using the−^1 notation. For example, ify=sinhx,then
x=arsinh yorx=sinh−^1 y. Similarly, ify=sechx,
thenx=arsechyorx=sech−^1 y,and so on.In this chap-
ter 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) Ify=sin−^1 x,thenx=siny.
Differentiating bothsides withrespect toygives:
dx
dy
=cosy=
√
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, wheny=sin−^1 xthen
dy
dx
=
1
√
1 −x^2
(ii) A sketch of part of the curve ofy=sin−^1 xis
shown in Fig. 33.1(a). The principal value of
sin−^1 x is defined as the value lying between
−π/2andπ/2.Thegradient of thecurvebetween
pointsAandBis positive for all values ofx
and thus only the positive value is taken when
evaluating
1
√
1 −x^2
(iii) Given y=sin−^1
x
a
then
x
a
=siny and
x=asiny
Hence
dx
dy
=acosy=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