Chain Rule: further examples

©Mathematics Support Centre,Coventry University, 2001
Examples
Differentiate the following:
t
ec
dt
dx
t
t
t
t
t
dt
dx
t
u
dt
dx
u
u
du
dx
t
dt
du
u
x
t
u
t
x
t
x
x
x
dx
dy
x
u
dx
dy
u
du
dy
x
v
v
dx
du
v
dv
du
dx
dv
v
u
x
v
u
u
y
x
u
x
y
x
x
dx
dy
x
u
dx
dy
u
du
dy
x
dx
du
u
y
x
u
x
x
x
y
5
15cos
5
cos
15
5
sin
5
cos
5
5sec
5
tan
3
5
5sec
3
3
3
1
5
5sec
3
tan5
)
3(tan5
Now
tan5
3
3.
7)
sin(3
7)
24(3
7)
12(3
2sin
2sin
7)
12(3
12
3
4
4
3
7
3
the chain rule
we must use
To differentiate
2cos
7)
(3
7)
2cos(3
2.
cos4
8sin 4
4cos4
2
2
4cos4
sin 4
let
)
(sin 4
4
Now sin
4
sin
1.
2
2
2
2
2
2
2
2
2
2
2
1
1
4
3
3
3
3
3
3
4
4
4
2
2
2
2
= −
×
= −
×
= −
×
= −
= −
×
= −
=
=
=
=
=
−
−
= −
−
×
= −
= −
−
=
=
×
=
=
=
=
−
=
=
−
=
−
=
=
×
=
=
=
=
=
=
=
−
−
−
Exercise
3
2
2
5
2
2
4
3
1
2
3
cos
4
10.
3)
ln(5
9.
)
ln(sin
8.
)
ln(sin
7.
)
4
2ln(3
6.
2
3cos
7)
(
5sin
5.
)
3
(
4.
3)
sin(2
1
3.
2
3tan
2.
cos
1.
x
y
x
y
t
x
t
x
q
p
t
t
s
x
e
y
x
y
x
y
x
y
x
=
+
=
=
=
−
=
+
−
=
+
=
+
=
=
=
−
Answers:
3
3
2
2
2
2
3
3
1
3
1
2
2
2
2
3
cos
sin
6
10.
3)
ln(5
3)
2(5
5
9.
2cot
sin
2cos
8.
cot
2
sin
cos
2
7.
)
4
(3
40
6.
cos2
12sin 2
7)
7) cos(
10sin(
5.
)
3
)(
12(1
4.
3)
(2
sin
3)
2cos(2
3.
2
sec
12 tan 2
2.
sin
3cos
1.
x
x
x
x
x
t
t
t
t
t
t
t
t
q
t
t
t
t
x
e
e
x
x
x
x
x
x
x
x
+
+
=
=
−
−
−
−
−
+
−
+
+
−
−
−
−