Multivariate Calculus - Bruce E. Shapiro

LECTURE 12. THE CHAIN RULE 87
because the velocity vector has been defined as v = r ′ (t). Thus
df
dt
= v(t) · ∇f = D v f(t)
where v = ‖v(t)‖ is the speed at time t. Thus the total rate of change of a function
as you move along a curve r(t) is the directional derivative in the direction of the
tangent (or velocity) vector.
Proof of the Chain Rule
We outline the proof for a function f (x(t), y(t)) of two variables; the generalization
to higher dimensions is the same.The derivative of f with respect to t is
df
dt = lim f(t + ∆t) − f(t)
∆t→0 ∆t
But
f(t + ∆t) = f(x(t + ∆t), y(t + ∆t))
so that
f(t + ∆t) − f(t) = f(x(t + ∆t), y(t + ∆t)) − f(x(t), y(t))
= f(x(t + ∆t), y(t + ∆t)) − f(x(t), y(t + ∆t))
+f(x(t), y(t + ∆t)) − f(x(t), y(t))
This is allowed because the two middle terms add to zero. Let
∆x = x(t + ∆t) − x(t), ∆y = y(t + ∆t) − y(t)
Then
f(t + ∆t) − f(t) = f(x + ∆x, y + ∆y) − f(x, y + ∆y)
+f(x, y + ∆y) − f(x(t), y(t))
Math 250, Fall 2006 Revised December 6, 2006.