Higher Engineering Mathematics, Sixth Edition

294 Higher Engineering Mathematics

Note that the differential coefficient isnotobtained by

merely differentiating each term in turn and then divid-

ing the numerator by the denominator. The quotient

formulamustbe used when differentiating quotients.

Problem 15. Determine the differential

coefficient ofy=tanax.

y=tanax=

sinax

cosax

. Differentiation of tanax is 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

,sincecos^2 ax+sin^2 ax= 1

(see Chapter 15)

Hence

dy

dx

=asec^2 ax since sec^2 ax=

1

cos^2 ax

(see

Chapter 11).

Problem 16. Find the derivative ofy=secax.

y=secax=

1

cosax

(i.e. a quotient). Letu=1and

v=cosax

dy

dx

=

v

du

dx

−u

dv

dx

v^2

=

(cosax)( 0 )−( 1 )(−asinax)

(cosax)^2

=

asinax

cos^2 ax

=a

(

1

cosax

)(

sinax

cosax

)

i.e.

dy

dx

=asecaxtanax

Problem 17. Differentiatey=

te^2 t

2cost

The function

te^2 t

2cost

is a quotient, whose numerator is a

product.

Letu=te^2 tandv=2costthen

du

dt

=(t)(2e^2 t)+(e^2 t)( 1 )and

dv

dt

=−2sint

Hence

dy

dx

=

v

du

dx

−u

dv

dx

v^2

=

(2cost)[2te^2 t+e^2 t]−(te^2 t)(−2sint)

(2cost)^2

=

4 te^2 tcost+2e^2 tcost+ 2 te^2 tsint

4cos^2 t

=

2e^2 t[2tcost+cost+tsint]

4cos^2 t

i.e.

dy

dx

=

e^2 t

2cos^2 t

(2tcost+cost+tsint)

Problem 18. Determine the gradient of the curve

y=

5 x

2 x^2 + 4

at the point

(

√

3 ,

√

3

2

)

.

Lety= 5 xandv= 2 x^2 + 4

dy

dx

=

v

du

dx

−u

dv

dx

v^2

=

( 2 x^2 + 4 )( 5 )−( 5 x)( 4 x)

( 2 x^2 + 4 )^2

=

10 x^2 + 20 − 20 x^2

( 2 x^2 + 4 )^2

=

20 − 10 x^2

( 2 x^2 + 4 )^2

At the point

(

√

3 ,

√

3

2

)

,x=

√

3,

hence the gradient=

dy

dx

=

20 − 10 (

√

3 )^2

[2(

√

3 )^2 +4]^2

=

20 − 30

100

=−

1

10

Now try the following exercise

Exercise 117 Further problemson

differentiating quotients

In Problems 1 to 7, differentiate the quotients with

respect to the variable.

1.

sinx

x

[

xcosx−sinx

x^2

]