Lecture 11 (Oct. 2)
Chain Rule (reading: 14.5)
Some versions of the chain rule we already know:
1) real-valued functions of a single variable: if g(t) is di↵erentiable at t = t0 , and
f (x) is di↵erentiable at x = g(t0 ), then f (g(t)) is di↵erentiable at t = t0 with
d
f (g(t))|t=t0 = f 0 (g(t0 ))g 0 (t0 )
dt
2) vector-valued functions of 1 variable: if s(t) (scalar function) is di↵erentiable
at t = t0 , and r(s) (vector function) is di↵erentiable at s = s(t), then r(s(t)) is
di↵erentiable at t = t0 with
d
r(s(t))|t=t0 = r0 (s(t0 ))s0 (t0 ).
dt
For functions of 2 variables, we have:
Chain Rule (version 1): suppose x(t) and y(t) are di↵erentiable at t = t0 , and suppose f (x, y) is di↵erentiable at (x, y) = (x(t0 ), y(t0 )). Then the composite function
f (x(t), y(t)) (a function of the single variable t) is di↵erentiable at t = t0 , with
d
f (x(t), y(t))|t=t0 = fx (x(t0 ), y(t0 ))x0 (t0 ) + fy (x(t0 ), y(t0 ))y 0 (t0 )
dt
✓
◆
@f dx @f dy
=
+
.
@x dt
@y dt
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Proof:
Example: An insect is moving in the plane. Its position at time t is given by x(t) =
cos(t), y(t) = 4 sin(t) (an ellipse!). The temperature at a point (x, y) in the plane is
T (x, y) = exy . What is the rate of change of temperature felt by the insect?
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Chain Rule (version 2): suppose x(s, t) and y(s, t) are di↵erentiable functions of s,
t, and suppose f (x, y) is a di↵erentiable function of x and y. Then the composite
function f (x(s, t), y(s, t)) is a di↵erentiable function of s and t, with
@
f (x(s, t), y(s, t)) =
@t
@
f (x(s, t), y(s, t)) =
@s
@f @x @f @y
+
@x @t
@y @t
@f @x @f @y
+
@x @s @y @s
Diagram for keeping track of chain rule:
Example: Let x = st, y = s2 t, z = st2 , f (x, y, z) = tan(xy) + z 2 . Find @f /@t.
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