Chain Rule, and Higher Derivatives

Chain Rule
We’ve got general procedures for differentiating expressions with addition, subtraction, and multiplication.
What about composition?
Example 1. y = f(x) = sin x,x = g(t)= t2.
So, y = f(g(t)) = sin(t2). To find dy , write
dt
t = t0 +?t
t0 = t0
x = x0 +?x
x0 = g(t0)
y = y0 +?y
y0 = f(x0)
?y =?y ?x ?t ?x · ?t As ?t 0, ?x 0 too, because of continuity. So we get:
??
dy dydx
= The Chain Rule!
dt
dx
dt ?
In the example,
dx dt
= 2t and
dy dx
= cos x.
So,
d dt
?sin(t2)?
=
( dy dx)(dx dt )
==
(cos x)(2t) (2t) ?cos(t2)?
Another notation for the chain rule
?
?
d dtf(g(t)) = f?(g(t))g?(t)
or
d dxf(g(x)) = f?(g(x))g?(x)
Example 1. (continued) Composition of functions f(x) = sin x and g(x)= x2 (fg)(x)= f(g(x)) = sin(x2)
?
(gf)(x)= g(f(x)) = sin2(x)
?
Note: f ? g ?= g ? f. Not Commutative!
Example 2. cos =?
dx x
1
Let u =
x
dy = dy du
dx du dx
dy du 1
du
= ?sin(u);
dx
= ?
x2 ? ?
? ? 1 sin
dy sin(u) x = = (?sin u) ?1
=
dx x2 x2 x2
d ? ?
Example 3. x?n =?
dx
? ?1 n 1
There are two ways to proceed. x?n = , or x?n =
x xn
1. d ?
x?n?
= d
?
1
?n
= n
?
1
?n?1 ?
?1
?
= ?nx?(n?1)x?2 = ?nx?n?1
dx dx x x x2
2. d ?
x?n?
= d 1
= nxn?1 ?1
= ?nx?n?1 (Think of xn as u)
dx dx xn x2n
2