Calculus Chapter 3

Derivatives

3.1 Informal definition of derivative

3.1 Informal definition of derivative

A derivative is a formula for the rate at which a function changes.

Formal Definitionof the Derivative of a function

You’ll need to “snow” this

Formal Definitionof the Derivative of a function

f’(x)= lim f(x+h) – f(x) h->0 h

Notation for derivative

y’ dy/dx df/dx d/dx (f) f’(x) D (f)

Rate of change and slope

Slope of a secant line

See diagram

The slope of the secant line gives the change between 2 distinct points on a

curve.

i.e. average rate of change

Rate of change and slope-slope of the tangent line to a curve

see diagram

The slope of the tangent line gives the rate of change at that one point

i.e. the instantaneous change.

compare Slope= y-y x-x Slope of secant line

m= f ’(x) Slope of tangent line

Time for examples Finding the derivative

using the formal definition

This is music to my ears!

A function has a derivative at a point

A function has a derivative at a point

iff the function’s right-hand and left-hand derivatives exist and are equal.

Theorem

If f (x) has a derivative at x=c,

Theorem

If f (x) has a derivative at x=c,

then f(x) is continuous at x=c.

Finding points where horizontal tangents to a curve occur

3.3 Differentiation Rules

1. Derivative of a constant

3.3 Differentiation Rules

1. Derivative of a constant

2. Power Rule for derivatives

3.3 Differentiation Rules

1. Derivative of a constant

2. Power Rule for derivatives

3. Derivative of a constant multiple

3.3 Differentiation Rules

1. Derivative of a constant

2. Power Rule for derivatives

3. Derivative of a constant multiple

4. Sum and difference rules

3.3 Differentiation Rules

1. Derivative of a constant

2. Power Rule for derivatives

3. Derivative of a constant multiple

4. Sum and difference rules

5. Higher order derivatives

3.3 Differentiation Rules

1. Derivative of a constant

2. Power Rule for derivatives

3. Derivative of a constant multiple

4. Sum and difference rules

5. Higher order derivatives

6. Product rule

3.3 Differentiation Rules

1. Derivative of a constant

2. Power Rule for derivatives

3. Derivative of a constant multiple

4. Sum and difference rules

5. Higher order derivatives

6. Product rule

7. Quotient rule

3.3 Differentiation Rules

1. Derivative of a constant

2. Power Rule for derivatives

3. Derivative of a constant multiple

4. Sum and difference rules

5. Higher order derivatives

6. Product rule

7. Quotient rule

8. Negative integer power rule

3.3 Differentiation Rules

1. Derivative of a constant2. Power Rule for derivatives3. Derivative of a constant multiple4. Sum and difference rules5. Higher order derivatives6. Product rule7. Quotient rule8. Negative integer power rule9. Rational power rule

3.4 Definition

Average velocity of a “body”

moving along a line

Defintion

Instantaneous Velocity

is the derivative of

the position function

Def. speed

Definition

Speed

The absolute value of velocity

Definition

Acceleration

acceleration Don’t drop the ball on this

one.

Definition

Acceleration

The derivative of velocity,

Definition

Acceleration

The derivative of velocity,

Also ,the second derivative of position

3.5 Derivatives of trig functions

Y= sin x

3.5 Derivatives of trig functions

Y= sin x Y= cos x

3.5 Derivatives of trig functions

Y= sin x Y= cos x Y= tan x

3.5 Derivatives of trig functions

Y= sin x Y= cos x Y= tan x Y= csc x

3.5 Derivatives of trig functions

Y= sin x Y= cos x Y= tan x Y= csc x Y= sec x

3.5 Derivatives of trig functions

Y= sin x Y= cos x Y= tan x Y= csc x Y= sec x Y= cot x

TEST 3.1-3.5 Formal def derivative Rules for derivatives Notation for derivatives Increasing/decreasing Eq of tangent line Position, vel, acc Graph of fct and der Anything else mentioned,

assigned or results of these

Whereas

The slope of the secant line gives the change between 2 distinct points on a

curve.

i.e. average rate of change