Higher Engineering Mathematics, Sixth Edition

324 Higher Engineering Mathematics

Whenx=2andy=3,

dy

dx

=

1 − 2

3 − 1

=

− 1

2

Hence the gradients of the tangents are±

1

2

The circle having the given equation has its centre at

(1, 1) and radius

√

5 (see Chapter 13) and is shown in

Fig. 30.2 with the two gradients of the tangents.

Gradient

521

2

Gradient

51

2

142 x

1

2

4

3

y

21

22

0

r^55

x 21 y 222 x

22 y 53

Figure 30.2

Problem 10. Pressurepand volumevof a gas

are related by the lawpvγ=k,whereγandkare

constants. Show that the rate of change of pressure

dp

dt

=−γ

p

v

dv

dt

Sincepvγ=k,thenp=

k

vγ

=kv−γ

dp

dt

=

dp

dv

×

dv

dt

by the function of a function rule

dp

dv

=

d

dv

(kv−γ)

=−γkv−γ−^1 =

−γk

vγ+^1

dp

dt

=

−γk

vγ+^1

×

dv

dt

Sincek=pvγ,

dp

dt

=

−γ(pvγ)

vγ+^1

dv

dt

=

−γpvγ

vγv^1

dv

dt

i.e.

dp

dt

=−γ

p

v

dv

dt

Now try the following exercise

Exercise 130 Further problems on implicit

differentiation

In Problems 1 and 2 determine

dy

dx

1. x^2 +y^2 + 4 x− 3 y+ 1 = 0

[

2 x+ 4

3 − 2 y

]

2. 2y^3 −y+ 3 x− 2 = 0

[

3

1 − 6 y^2

]

3. Givenx^2 +y^2 =9evaluate

dy

dx

when

x=

√

5andy=2.

[

−

√

5

2

]

In Problems 4 to 7, determine

dy

dx

4. x^2 + 2 xsin4y= 0

[

−(x+sin4y)

4 xcos4y

]

5. 3y^2 + 2 xy− 4 x^2 = 0

[

4 x−y

3 y+x

]

6. 2x^2 y+ 3 x^3 =siny

[

x( 4 y+ 9 x)

cosy− 2 x^2

]

7. 3y+ 2 xlny=y^4 +x

[

1 −2lny

3 +( 2 x/y)− 4 y^3

]

8. If 3x^2 + 2 x^2 y^3 −

5

4

y^2 =0evaluate

dy

dx

when

x=

1

2

andy=1. [5]

9. Determine the gradients of the tangents
   drawn to the circlex^2 +y^2 =16 at the point
   wherex=2. Give the answer correct to 4
   significant figures. [± 0 .5774]


10. Find the gradients of the tangents drawn to

the ellipse

x^2

4

+

y^2

9

=2 at the point where

x=2. [± 1 .5]

11. Determine the gradient of the curve
    3 xy+y^2 =−2 at the point (1,−2). [−6]