§5. Differentiable functions of several variables
279
One may also observe that the formulae (15.8)
d(f+g) =dl+dg, dUg) =gdl+ldg, d(l/!) = -dfll 2
extend to functions of several variables; it is then clear that if the differentials
of I and 9 are proportional to dz, then, similarly, so are those of I + g, Ig
and I/g·
Exercise. Write a polynomial function P in the form
calculate 8P/8z and 8P/8z.
21 - Differentiation of composite functions
The formula for differentiating composite functions (the Chain Rule) generalises without a problem - apart from the notation - to Cl functions. First
consider the simplest, and very useful, case: we have a function
of class Cion an open V c IC and a differentiable map t I-> I(t) =
(!I(t), h(t)) of an interval or, more generally, of an open subset 42 U of lR
into V; we may then consider the composite function
pet) = go I(t) = g[/(t)] = g[!I (t), h(t)J,
with values in IC. For t given and h E lR such that t + h E U let us put
!I(t + h) = !I(t) + u,
h(t + h) = h(t) + v,
whence
u = I~ (t)h + o(h) = ah + o(h),
v = I~(t)h + o(h) = bh + o(h).
Since 9 is differentiable at the point I(t), with partial derivatives which we
denote by c and d, it follows that
pet + h)
pet) + cu + dv + o(u) + o(v) =
pet) + (ca + db)h + o(h)
42 Since an open set in R is, on a neighbourhood of each of its points, identical to
an interval, everything that applies to functions defined on an interval applies to
the general case.