1. C3 DIFFERENTIATION Answers - Worksheet I
1 −1
1 a 2
y 2
b y = x2
c 2x
dx 1 −1 1 1
d = 2
y 2
= =
dy 2 y 2x
1 1 dy 1
= = 2x ∴ =
( dy )
d
x ( 21x ) dx ( dy )
d
x
dy dy dx −1 1 1
2 a = 2e2x − 1 b = 3x2 c = 1
2
(ln y) 2 × =
dx dx dy y 2 y ln y
1 2
x= 1
2
(ln y + 1) x = ( y − 2) 3 y = ex
dx 1 1 dx −2 1 dy 2
= = 2 x−1
= 1 ( y − 2)
3
3
= = 2 xe x = 2 y ln y
dy 2y 2e dy 3x 2 dx
dy dx 1 dy dx 1 dy dx 1
× = 2e2x − 1× 2 x−1
=1 × = 3x2 × =1 × = 2 y ln y × =1
dx dy 2e dx dy 3x 2 dx dy 2 y ln y
dx dx dx
3 a = 2y b = 3(y − 1)2 × 1 c = sec2 y
dy dy dy
dy 1 dy 1 dy
∴ = ∴ = ∴ = cos2 y
dx 2y dx 3( y − 1)2 dx
dx 1 dx dx 1 × e y − ( y − 2) × e y 3− y
d = ×3 e = 2 sin y cos y = sin 2y f = =
dy 3y + 2 dy dy (e y ) 2 ey
dy 3y + 2 dy dy ey
∴ = ∴ = cosec 2y ∴ =
dx 3 dx dx 3− y
dx
4 a = 3y2 − 8y 5 a ey = ax + b
dy
1
b y = 3 ∴ x = −9 x= (ey − b)
a
dx dx 1
=3 b = ey
dy dy a
dy d dy dx
∴ grad = = 1
3
c [ln (ax + b)] = =1÷
dx dx dx dy
a a
∴ y−3= 1
3
(x + 9) = =
ey ax + b
1
[y= 3
x+6]
6 a ln y = ln 3x = x ln 3
ln y
∴ x=
ln 3
dx 1 1 1
b = × =
dy ln 3 y y ln 3
dy dx
c =1÷ = y ln 3
dx dy
= 3x ln 3
d grad = 9 ln 3
∴ y − 9 = (9 ln 3)(x − 2)
[ y = 9x ln 3 + 9 − 18 ln 3 ]
 Solomon Press