Higher Engineering Mathematics

METHODS OF DIFFERENTIATION 293

G

Now try the following exercise.

Exercise 118 Further problems on differen-

tiating products

In Problems 1 to 5 differentiate the given prod-

ucts with respect to the variable.

1. 2x^3 cos 3x [6x^2 ( cos 3x−xsin 3x)]

2.

√

x^3 ln 3x

[√

x

(

1 +^32 ln 3x

)]

3. e^3 tsin 4t [e^3 t(4 cos 4t+3 sin 4t)]


4. e^4 θln 3θ

[

e^4 θ

(

1

θ

+4ln3θ

)]

5. etlntcost
   [
   et

{(

1

t

+lnt

)

cost−lntsint

}]

6. Evaluate

di

dt

, correct to 4 significant figures,

whent= 0 .1, andi= 15 tsin 3t.

[8.732]

7. Evaluate

dz

dt

, correct to 4 significant figures,

whent= 0 .5, given thatz=2e^3 tsin 2t.

[32.31]

27.5 Differentiation of a quotient

Wheny=

u

v

, anduandvare both functions ofx

then

dy

dx

=

v

du

dx

−u

dv

dx

v^2

This is known as thequotient rule.

Problem 14. Find the differential coefficient of

y=

4 sin 5x

5 x^4

.

4 sin 5x

5 x^4

is a quotient. Letu=4 sin 5xandv= 5 x^4

(Note thatvisalwaysthe denominator anduthe

numerator)

dy

dx

=

v

du

dx

−u

dv

dx

v^2

where

du

dx

=(4)(5) cos 5x=20 cos 5x

and

dv

dx

=(5)(4)x^3 = 20 x^3

Hence

dy

dx

=

(5x^4 )(20 cos 5x)−(4 sin 5x)(20x^3 )

(5x^4 )^2

=

100 x^4 cos 5x− 80 x^3 sin 5x

25 x^8

=

20 x^3 [5xcos 5x−4 sin 5x]

25 x^8

i.e.

dy

dx

=

4

5 x^5

(5xcos 5x−4 sin 5x)

Note that the differential coefficient isnotobtained

by merely differentiating each term in turn and then

dividing the numerator by the denominator. The quo-

tient formulamustbe used when differentiating

quotients.

Problem 15. Determine the differential coeffi-

cient ofy=tanax.

y=tanax=

sinax

cosax

. Differentiation of tanaxis thus

treated as a quotient withu=sinaxandv=cosax

dy

dx

=

v

du

dx

−u

dv

dx

v^2

=

(cosax)(acosax)−(sinax)(−asinax)

(cosax)^2

=

acos^2 ax+asin^2 ax

( cosax)^2

=

a(cos^2 ax+sin^2 ax)

cos^2 ax

=

a

cos^2 ax

, since cos^2 ax+sin^2 ax= 1

(see Chapter 16)

Hence

dy

dx

=asec^2 ax since sec^2 ax=

1

cos^2 ax

(see Chapter 12).