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Past exam of the mathematics course of the University of Cambridge
/
2024
/
ia
/
Paper 1
/
10E
/
c
/
i
/
Solution
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...
2024
ia
Paper 1
10E
c
i
Created
2026-09-24
Updated
2026-09-25
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Differentiability lets us write
f
(
a
+
h
)
=
f
(
a
)
+
f
′
(
a
)
h
+
r
(
h
)
,
h
r
(
h
)
→
0
,
(1)
and, with
k
→
0
,
g
(
f
(
a
)
+
k
)
=
g
(
f
(
a
))
+
g
′
(
f
(
a
))
k
+
s
(
k
)
,
k
s
(
k
)
→
0.
(2)
Substitute
k
=
f
(
a
+
h
)
−
f
(
a
)
=
f
′
(
a
)
h
+
r
(
h
)
. Since
k
=
O
(
h
)
, the
remainder
s
(
k
)
is
o
(
h
)
. Thus
g
(
f
(
a
+
h
))
−
g
(
f
(
a
))
=
g
′
(
f
(
a
))
f
′
(
a
)
h
+
o
(
h
)
,
(3)
and the
chain rule
is
(
g
∘
f
)
′
(
a
)
=
g
′
(
f
(
a
))
f
′
(
a
)
.
(4)
Ancestors
(12)
i
c
10E
Paper 1
ia
2024
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
List of universities
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