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