3.6 The Chain Rule
The Rule
Rule
dy du
dy
=
dx
du dx
The Rule
Rule
dy du
dy
=
dx
du dx
or
Rule
d
d
(f g)(x) = f (g(x) = f 0 (g(x) g0 (x)
dx
dx
The Rule
Rule
dy du
dy
=
dx
du dx
or
Rule
d
d
(f g)(x) = f (g(x) = f 0 (g(x) g0 (x)
dx
dx
or
Rule
d
du
f (u) =
3.3 Differentiation Rules
Constant Functions
What does a constant function look like?
Constant Functions
What does a constant function look like?
y
y=2
x
Constant Functions
What does a constant function look like?
y
y=2
x
What is the slope?
Constant Func
3.7 Implicit Differentiation
Explicit v. Implicit Differentiation
Every example we have done so far has been explicit differentiation.
We will now look at implicit differentiation.
Explicit v. Implicit Differentiation
Every example we have done so far ha
3.11 Linearization and Differentials
Local Linearization
Local linearization, or tangent line approximations, is the use of
tangent lines to approximate the value of a function at a point.
Local Linearization
Local linearization, or tangent line approxim
3.2 The Derivative as a Function
The Definition
We have done this before .
The Definition
We have done this before .
Definition
The derivative of the function f (x) with respect to the variable x is the
function f 0 whose value at x is
f 0 (x) = lim
h0
p
3.4 The Derivative as a Rate of Change
Useful Derivatives
Speed, Velocity and Acceleration
If s(t) is the position at time t, then
v(t) =
ds
dt
Useful Derivatives
Speed, Velocity and Acceleration
If s(t) is the position at time t, then
v(t) =
ds
dt
Spe
3.8 Derivatives of Inverse Functions and
Logarithms
Derivative of f and f 1
y = 2x + 1
Derivative of f and f 1
y = 2x + 1
y0 = 2
Derivative of f and f 1
y = 2x + 1
y0 = 2
y=
x1
2
Derivative of f and f 1
y = 2x + 1
y0 = 2
y = x1
2
y0 = 12
Derivative of f
3.5 Derivatives of Trigonometric
Functions
Starting with the Sine Function
f (x) = sin x
Starting with the Sine Function
f (x) = sin x
Starting with the Sine Function
f (x) = sin x
Starting with the Sine Function
f (x) = sin x
Starting with the Sine Func
Steps for Curve Sketching
1. Find f 0 (x) and f 00 (x).
2. Find all relative extrema points by setting f 0 (x) = 0 and then substitute each solution into
f (x).
3. Substitute each relative extrema point found in step 2 into f 00 (x) to classify the critic
Proof of Logarithmic Rules
Rule 1: The Power Rule
loga xn = n loga x
Proof : Let m = loga x. Then, using the definition of logarithms, we can rewrite this as
m = loga x x = am
Now,
x = am
xn = (am )n
Writing back in logarithmic form and substituting, we h
1.3 Trigonometric Functions
Angles
Who can define an angle?
Angles
Who can define an angle?
Definition
An angle is a shape formed by two lines or rays diverging to a
common point.
Angles
Who can define an angle?
Definition
An angle is a shape formed by t
1.1 Functions and Their Graphs
What is a function?
What is a function?
What is a function?
What is a function?
Can we formally define a function?
What is a function?
What is a function?
Can we formally define a function?
Definition
A function is a relati
2.6 Limits Involving Infinity: Asymptotes
of Graphs
What We Will Look At
In this section, we will look at two main ideas:
What We Will Look At
In this section, we will look at two main ideas:
1
Limits at infinity
What We Will Look At
In this section, we
1.2 Combining Functions: Shifting and
Scaling Graphs
Adding and Subtracting Functions
Example
If f (x) = x2 and g(x) = 2x2 + 3, what is (f + g)(x)?
Adding and Subtracting Functions
Example
If f (x) = x2 and g(x) = 2x2 + 3, what is (f + g)(x)?
(f + g)(x)
2.4 One-Sided Limits
One-Sided Limits
f (x) =
x
|x|
Definitions
Definition
The left-hand limit lim f (x) means we are approaching c through
xc
values less than c, or from the left.
Definitions
Definition
The left-hand limit lim f (x) means we are approac
2.1 Rates of Change and Tangents to
Curves
Our Goal
a
x1
Our Goal
f (x1 )
f (a)
a
x1
Our Goal
f (x1 )
f (a)
a
x1
Our Goal
f (x1 )
f (x2 )
f (a)
a
x2
x1
Our Goal
f (x1 )
f (x2 )
f (a)
a
x2
x1
Our Goal
f (x1 )
f (a)
a
f (x2 )
f (x3 )
x3
x2
Our Goal
f (x1 )
3.1 Tangents and the Derivative at a Point
Formal Definition
Definition
The derivative of a function at a point x0 , denoted f 0 (x0 ), is
f 0 (x0 ) = lim
h0
provided the limit exists.
f (x0 + h) f (x0 )
h
Formal Definition
Definition
The derivative of a
2.2 Limit of a Function and Limit Laws
How Limits Work
Example
2
4
Find lim xx2
x2
How Limits Work
Example
2
4
Find lim xx2
x2
Definition of Limits
The Limit
Let f (x) be defined on an open interval about c, except possibly at c
itself. If f (x) is arbit
2.5 Continuity
Continuity - An Example
Example
Is the given function continuous?
Continuity - An Example
Example
Is the given function continuous?
Continuity - An Example
Example
Is the given function continuous?
Continuity - An Example
So, the function
2.3 The Precise Definition of the Limit
Definition of Limits
Precise Definition
Let f (x) be a function defined on an interval that contains x = c,
except possibly at x = c. Then,
lim f (x) = L
xc
if for every number > 0, there is some number > 0 such th