Higher Engineering Mathematics

Differential calculus

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 201).

Inverse trigonometric functionsare denoted by

prefixing the function with ‘arc’ or, more com-

monly, by using the−^1 notation. For example, if

y=sinx, thenx=arcsin y orx=sin−^1 y. Similarly,

ify=cosx, thenx=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=arsinhyorx=sinh−^1 y. Similarly, ify=sechx,

thenx=arsechyorx=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) Ify=sin−^1 x, thenx=siny.

Differentiating both sides with respect toy

gives:

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 x

is shown in Fig. 33(a). The principal value of

sin−^1 xis defined as the value lying between

−π/2 and π/2. The gradient of the curve

between pointsAandBis positive for all values

ofxand 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

Thus

dy

dx

=

1

dx

dy

=

1

√

a^2 −x^2

i.e.wheny=sin−^1

x

a

then

dy

dx

=

1

√

a^2 −x^2

Since integration is the reverse process of

differentiation then:

∫

1

√

a^2 −x^2

dx=sin−^1

x

a

+c

(iv) Giveny=sin−^1 f(x) the function of a function

rule may be used to find

dy

dx

.