DIFFERENTIATION OF IMPLICIT FUNCTIONS 323

G

Gradient

(^0124) x
Gradient
r^ = √
5
− 1
− 2
3
2
1
y
4
x^2 +y^2 − 2 x− 2 y= 3
=^12
= −^12
Figure 30.2
The circle having the given equation has its centre at
(1, 1) and radius
√
5 (see Chapter 14) and is shown
in Fig. 30.2 with the two gradients of the tangents.
Problem 10. Pressurepand volumevofagas
are related by the lawpvγ=k, whereγandk
are 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
Since k=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 132 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 =9 evaluate

dy

dx

when

x=

√

5 andy= 2

[

−

√

5

2

]

In Problems 4 to 7, determine

dy

dx

4.x^2 + 2 xsin 4y= 0

[

−(x+sin 4y)

4 xcos 4y

]

5. 3y^2 + 2 xy− 4 x^2 = 0

[

4 x−y

3 y+x

]

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

[

x(4y+ 9 x)

cosy− 2 x^2

]

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

[

1 −2lny

3 +(2x/y)− 4 y^3

]

8. If 3x^2 + 2 x^2 y^3 −

5

4

y^2 =0 evaluate

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]