Calculus Chapter 3 Derivatives

3.1 Informal definition of derivative

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

Formal Definition of the Derivative of a function

 You’ll need to “snow” this

Formal Definition of 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

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 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 9. 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  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