C3 Differentiation - Implicit differentiation
1.
PhysicsAndMathsTutor.com
A curve has equation 7x2 + 48xy – 7y2 + 75 = 0.
A and B are two distinct points on the curve. At each of these points the gradient of the curve
is equal to 112 .
(a)
Use implicit differentiation to show that x + 2y = 0 at the points A and B.
(5)
(b)
Find the coordinates of the points A and B.
(4)
(Total 9 marks)
2.
Differentiate with respect to x
(i)
x3 e3x,
(3)
(ii)
2x
,
cos x
(3)
(iii)
tan2 x.
(2)
Given that x = cos y2,
(iv)
find
dy
in terms of y.
dx
(4)
(Total 12 marks)
Edexcel Internal Review
1
C3 Differentiation - Implicit differentiation
1.
(a)
14x + (48x
dy
dy
+ 48y) – 14y
=0
dx
dx
dy 2
=
into derived expression to obtain
dx 11
96
28
14x +
x + 48y –
y=0
11
11
∴250x + 500y = 0 ⇒ x + 2y = 0
Substitutes
(b)
Eliminates one variable to obtain, for example,
7(2y)2 + 48(–2y)y – 7y2 + 75 = 0 and obtains y (or x)
Substitutes y to obtain x (or y)
Obtains coordinates (–2,1) and (2,–1)
PhysicsAndMathsTutor.com
M1 (B1) A1
M1
A1
5
M1
A1, A1
4
[9]
2.
(i)
u = x3
du
= 3x2
dx
dv
= 3e3x
dx
v = e3x
dy
= 3x2 e3x + x33e3x or equiv
dx
(ii)
u = 2x
3
M1 A1 A1
3
du
=2
dx
v = cos x
dv
= –sin x
dx
dy
2 cos x + 2 x sin x
or equiv
=
cos 2 x
dx
(iii)
M1 A1 A1
u = tan x
du
= sec2 x
dx
y = u2
dy
= 2u
du
dy
= 2u sec2 x
dx
dy
= 2 tan x sec2 x
dx
Edexcel Internal Review
M1
A1
2
2
C3 Differentiation - Implicit differentiation
(iv)
u = y2
du
= 2y
dy
x = cos u
dx
= − sin u
du
dx
= − 2y sin y2
dy
−1
dy
=
dx 2 y sin y 2
PhysicsAndMathsTutor.com
M1
A1
M1 A1
4
[12]
Edexcel Internal Review
3