Differentiation: Chain Rule For Composite Functions
Recommendation: Learn how to apply the chain rule first. Leave the
challenging parts of the following proof to later.
The following proof is based on
the continuity of composite functions and based on
properties of linear
approximation. The latter also imply a theorem on the existence and
calculation of derivatives of inverse functions. See one of the next
webpages.
Theorem (Chain Rule): Suppose z = f (y) is
differentiable at y = b, so that f '(b) is a real number. Suppose y =
g(x) is differentiable at x = a with g(a) = b. Then z = f(g(x)) =
h(x) is differentiable at x = a with
h ' (a) = f '(b) g'(a)
The proof given here is a simplification of the little o and
big (letter) O arguments which I first saw in the Loomis and Sternberg
textbook Advanced Calculus. Calculus textbooks, those I
read as a student took a more complicated route:
In the 1960s, most proofs of the chain rule followed a
different route that that given below. In it, the possibility of division by
zero needed to be considered. The Linear approximation based proof avoids the
possibility, and shows how a different starting point may ease or lessen the
difficulty of a logical development of a proof , and more generally college
and high school mathematics.
Proof: By the linear approximation theorem
f(y) - f(b) = f '(b) (y-b) + Rb(y)(y-b)
for some function Rb(y) with Rb(b) = 0 = lim y® b Rb(y). Now y =
g(x) and h(x) = f(g(x)) = f(y) gives
g(x) - g(a)
= f(y) - f(g(a)) = f(y) - f(b) = f '(b) (y-b) +
Rb(y)(y-b)
Therefore
|
h(x) - h(a)
x- a |
= |
f '(b) (y-b) + Rb(y)(y-b)
x-a |
|
= |
f '(b) (y-b)
x-a |
+ |
Rb(y)(y-b)
x-a |
|
= |
f '(b) |
y-b
x-a |
+ |
Rb(y) |
y-b
x-a |
|
= |
f '(b) |
g(x)-g(a)
x-a |
+ |
Rb(y) |
g(x)-g(a)
x-a |
Therefore
|
h'(a) |
= |
lim
x ® a |
h(x) - h(a)
x- a |
|
|
= |
lim
x ® a |
[f '(b) . |
g(x)-g(a)
x-a |
+ |
Rb(y). |
g(x)-g(a)
x-a |
] |
|
= |
|
f '(b) . g'(a) |
+ |
0 . g'(a) |
|
= |
|
f '(b) . g'(a) |
|
|
as required.
| |
Calculus Appetizers
& Lessons
Starter Guide (Views)
Real Player Videos
3. Derivative Motivation
3. Derivative Definition I
3. Derivatives Definition II
3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x) (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
3. Product Rule
3. Power Rule
3. Previous Rules Combined
3. d/dx for Polynomials
3. Reciprocal Rule
3. Reciprocal Law: sec & csc
3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
3. Quotient Rule Examples
3. Quotient Rule: tan & cot
3. Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots
YOU are better than YOU think. Show yourself how:
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Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
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Caution: Site advice is approximately
correct, for some circumstances, not all. . That leaves room for thought and
refinement.. |
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