ABSTRACT
The derivative rules from first-term calculus can be written in differen-
tial form as
d(un) = nun−1 du, (11.2)
d(eu) = eu du, (11.3)
d(sin u) = cosu du, (11.4)
d(uv) = u dv + v du, (11.5)
d(u+ cv) = du+ c dv, (11.6)
where c is a constant. The first three rules suffice to differentiate all ele-
mentary functions,1 and the latter two, referred to as the Leibniz property
and linearity, respectively, are distinguishing characteristics of many differ-
ent notions of differentiation. The quotient rule is not needed, as it follows
from the product rule together with the power rule. Derivatives of inverse
functions are most easily computed directly, for example,
u = ln v =⇒ v = eu =⇒ dv = eu du = v du =⇒ du = dv v . (11.7)
Finally, the chain rule is conspicuous by its absence. For example, if
f = q2 and q = sinu, then
df = 2q dq = 2 sinu cosu du (11.8)
so that df
du =
df
dq
dq
du (11.9)
so that the traditional chain rule is built into differential notation. Put
differently, we have
df = df
dq dq =
df
du du. (11.10)
The differential df is not itself a derivative, as we have not yet specified
“with respect to what.” Rather, the derivative of f with respect to, say, u
is just the ratio of the small changes df and du.
