322 DIFFERENTIAL CALCULUS

x^2 +y^2 =25 is the equation of a circle, centre at
the origin and radius 5, as shown in Fig. 30.1. At
x=4, the two gradients are shown.

y

5

3

0

− 3

− 5

− 5 4 5 x

Gradient

= −^43

Gradient

=^43

x^2 + y^2 = 25

Figure 30.1

Above,x^2 +y^2 =25 was differentiated implicitly;
actually, the equation could be transposed to

y=

√
(25−x^2 ) and differentiated using the function
of a function rule. This gives

dy

dx

=

1

2

(25−x^2 )

− 1

(^2) (− 2 x)=−
x
√
(25−x^2 )
and when x=4,
dy
dx
=−
4
√
(25− 42 )
=±
4
3
as
obtained above.
Problem 8.
(a) Find
dy
dx
in terms ofxandygiven
4 x^2 + 2 xy^3 − 5 y^2 =0.
(b) Evaluate
dy
dx
whenx=1 andy=2.
(a) Differentiating each term in turn with respect to
xgives:
d
dx
(4x^2 )+
d
dx
(2xy^3 )−
d
dx
(5y^2 )=
d
dx
(0)
i.e. 8x+
[
(2x)
(
3 y^2
dy
dx
)
+(y^3 )(2)
]
− 10 y
dy
dx
= 0
i.e. 8 x+ 6 xy^2
dy
dx

• 
  

  2 y^3 − 10 y
  dy
  dx
  = 0
  Rearranging gives:
  8 x+ 2 y^3 =(10y− 6 xy^2 )
  dy
  dx
  and
  dy
  dx

  
  

  8 x+ 2 y^3
  10 y− 6 xy^2

  
  

  4 x+y^3
  y(5− 3 xy)
  (b) Whenx=1 andy=2,
  dy
  dx

  
  

  4(1)+(2)^3
  2[5−(3)(1)(2)]

  
  

  12
  − 2
  =− 6
  Problem 9. Find the gradients of the tangents
  drawn to the circle x^2 +y^2 − 2 x− 2 y=3at
  x=2.
  The gradient of the tangent is given by
  dy
  dx
  Differentiating each term in turn with respect tox
  gives:
  d
  dx
  (x^2 )+
  d
  dx
  (y^2 )−
  d
  dx
  (2x)−
  d
  dx
  (2y)=
  d
  dx
  (3)
  i.e. 2 x+ 2 y
  dy
  dx
  − 2 − 2
  dy
  dx
  = 0
  Hence (2y−2)
  dy
  dx
  = 2 − 2 x,
  from which
  dy
  dx

  
  

  2 − 2 x
  2 y− 2

  
  

  1 −x
  y− 1
  The value ofywhenx=2 is determined from the
  original equation
  Hence (2)^2 +y^2 −2(2)− 2 y= 3
  i.e. 4 +y^2 − 4 − 2 y= 3
  or y^2 − 2 y− 3 = 0
  Factorising gives: (y+1)(y−3)=0, from which
  y=−1ory= 3
  Whenx=2 andy=−1,
  dy
  dx

  
  

  1 −x
  y− 1

  
  

  1 − 2
  − 1 − 1

  
  

  − 1
  − 2

  
  

  1
  2
  Whenx=2 andy=3,
  dy
  dx

  
  

  1 − 2
  3 − 1

  
  

  − 1
  2
  Hence the gradients of the tangents are±
  1
  2