1Differentiation
IA Differential Equations
1.3 Methods of differentiation
Theorem (Chain rule). Given f(x) = F (g(x)), then
df
dx
=
dF
dg
dg
dx
.
Proof. Assuming that
dg
dx
exists and is therefore finite, we have
df
dx
= lim
h→0
F (g(x + h)) − F (g(x))
h
= lim
h→0
F [g(x) + hg
0
(x) + o(h)] − F (g(x))
h
= lim
h→0
F (g(x)) + (hg
0
(x) + o(h))F
0
(g(x)) + o(hg
0
(x) + o(h)) − F (g(x))
h
= lim
h→0
g
0
(x)F
0
(g(x)) +
o(h)
h
= g
0
(x)F
0
(g(x))
=
dF
dg
dg
dx
Theorem (Product Rule). Give f(x) = u(x)v(x). Then
f
0
(x) = u
0
(x)v(x) + u(x)v
0
(x).
Theorem (Leibniz’s Rule). Given f = uv, then
f
(n)
(x) =
n
X
r=0
n
r
u
(r)
v
(n−r)
,
where f
(n)
is the n-th derivative of f.