Notation The following notation is used Differentiation rules In the following a is
a constant and u, v and f are functions of
interchangeably to denote derivatives:
df
d2 f
Leibniz’s (1675) notation: dx and dx2 for x.
the first and second derivative.
f=
f0 =
Lagrange’s (1772) notation: f 0 , f 00 for
the first and second derivative. This notaa
0
tion is shorter but assumes that we know
au
au0
that derivates are taken with respect to x.
u+v
u0 + v 0
Taking the first derivative of f 0 we obtain
uv
u0 v + uv 0
the second derivative f 00 .
u(v)
u0 (v)v 0
Note that we write f and f (x) interu0 v − v 0 u
u
changeably. The former is shorter, but it
v
v2
assumes that we know that f is actually a
function of x.
xn
nxn−1
Optimisation
ex
ex
ln x
1/x
• Minimum: f 0 (x) = 0 and f 00 (x) > 0
sin x
cos x
• Maximum: f 0 (x) = 0 and f 00 (x) < 0
cos x
− sin x
1