CHAPTER
4
4.1
4.2
4.3
4.4
CHAPTER
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
CHAPTER
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
CHAPTER
7
7.1
7.2
7.3
7.4
7.5
CHAPTER
8
8.1
8.2
8.3
8.4
8.5
Contents
The
Chain
Rule
Derivatives by the Chain Rule
Implicit Differentiation and Related Rates
Inverse Functions and Their Derivatives
Inverses of Trigonometric Functions
Integrals
The Idea of the Integral
177
Antiderivatives
182
Summation vs. Integration
187
Indefinite Integrals and Substitutions
195
The Definite Integral
201
Properties of the Integral and the Average Value
206
The Fundamental Theorem and Its Consequences
213
Numerical Integration
220
Exponentials
and
Logarithms
An Overview
228
The Exponential
ex
236
Growth and Decay in Science and Economics
242
Logarithms
252
Separable Equations Including the Logistic Equation
259
Powers Instead of Exponentials
267
Hyperbolic Functions
277
Techniques
of Integration
Integration by Parts
Trigonometric Integrals
Trigonometric Substitutions
Partial Fractions
Improper Integrals
Applications
of
the
Integral
Areas and Volumes by Slices
Length of a Plane Curve
Area of a Surface of Revolution
Probability and Calculus
Masses and Moments
8.6
Force, Work, and Energy
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
1
C H A P T E R
4
Derivatives
by
the Chain Rule
1
4.1
The
Chain
Rule
You remember that the derivative of f(x)g(x) is not (df/dx)(dg/dx). The derivative
of sin x times x2 is not cos x times 2x. The product rule gave two terms, not one
term. But there is another way of combining the sine function
f
and the squaring
function g into a single function. The derivative of that new function does involve
the cosine times 2x (but with a certain twist). We will first explain the new function,
and then find the "chain rule" for its derivative.
May
I
say here that the chain rule is important. It is easy to learn, and you will
use it often. I see it as the third basic way to find derivatives of new functions from
derivatives of old functions. (So far the old functions are xn, sin x, and cos x. Still
ahead are
ex
and log x.) When
f
and g are added and multiplied, derivatives come
from the
sum rule
and
product rule.
This section combines
f
and g in a third way.
The new function is sin(x2)the
sine of x2. It is created out of the two original
functions: if x
=
3
then x2
=
9
and sin(x2)
=
sin
9.
There is a "chain" of functions,
combining sin x and x2 into the composite function sin(x2). You start with x,
then
find
g(x), then Jindf (g(x)):
The squaring function gives
y
=
x2. This is g(x).
The sine function produces
z
=
sin y
=
sin(x2).This is f(g(x)).
The "inside function" g(x) gives y.
This is the input to the
"outside function" f(y). That
is called composition. It starts with x and ends with
z.
The composite function is
sometimes written fog (the circle shows the difference from an ordinary product fg).
More often you will see f(g(x)):
Other examples are cos 2x and (
2
~
)
~
,
with g
=
2x.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 PETRINA
 Chain Rule, Derivative, Sin, Inverse function

Click to edit the document details