Basic Engineering Mathematics, Fifth Edition

318 Basic Engineering Mathematics

5. y=

1

x

6. y= 12


7. y=x−

1

x^2

8. y= 3 x^5 − 2 x^4 + 5 x^3 +x^2 − 1


9. y=

2

x^3

10. y= 4 x( 1 −x)


11. y=

√

x 12. y=

√

t^3

13. y= 6 +

1

x^3

14. y= 3 x−

1

√

x

+

1

x

15. y=(x+ 1 )^2 16. y=x+ 3

√

x

17. y=( 1 −x)^2 18. y=

5

x^2

−

1

√

x^7

+ 2

19. y= 3 (t− 2 )^2 20. y=

(x+ 2 )^2

x

21. Find the gradient of the following curves at
    the given points.
    (a) y= 3 x^2 atx= 1
    (b) y=

√

xatx= 9

(c) y=x^3 + 3 x−7atx= 0

(d) y=

1

√

x

atx= 4

(e) y=

1

x

atx= 2

(f) y=( 2 x+ 3 )(x− 1 )atx=− 2

22. Differentiate f(x)= 6 x^2 − 3 x+5andfind
    the gradient of the curve at
    (a) x=−1(b)x= 2


23. Find the differential coefficient of
    y= 2 x^3 + 3 x^2 − 4 x−1 and determine the
    gradient of the curve atx=2.


24. Determine the derivative of
    y=− 2 x^3 + 4 x+7 and determine the
    gradient of the curve atx=− 1. 5

34.6 Differentiation of sine and cosine functions

Figure 34.5(a) shows a graph ofy=sinx. The gradient

is continually changing as the curve moves from 0 to

y

0

(a)

(b)

0

y 5 sinx

x radians

x radians

1

2

2

A

A 9

09

C 9

B 9

D 9

BD

C



2

d

dx

1 dydx

2 

2 

3 

2



 3 

2

(sinx) 5 cosx



2

Figure 34.5

AtoBtoCtoD. The gradient, given by

dy

dx

,maybe

plotted in a corresponding position belowy=sinx,as

shown in Figure 34.5(b).

At 0, the gradient is positiveand is at its steepest. Hence,

0 ′is a maximum positive value. Between 0 andAthe

gradient is positive but is decreasing in value until atA

the gradient is zero, shown asA′. BetweenAandBthe

gradient is negative but is increasing in value until atB

the gradient is at its steepest. HenceB′is a maximum

negative value.

If the gradient of y=sinx is further investigated

betweenBandCandCandDthen the resulting graph

of

dy

dx

is seen to be acosine wave.

Hence the rate of change of sinxis cosx,i.e.

ify=sinxthen

dy

dx

=cosx

It may also be shown that

ify=sinax,

dy

dx

=acosax (1)

(whereaisaconstant)

andify=sin(ax+α),

dy

dx

=acos(ax+α) (2)

(whereaandαare constants).

If a similar exercise is followed fory=cosxthen the

graphs of Figure 34.6 result, showing

dy

dx

to be a graph

of sinxbut displaced byπradians.