Calculus derivatives

7/28/2019 Calculus derivatives

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Calculus Chapter 3

Derivatives



3.1 Informal definition of derivative



3.1 Informal definition of derivativeA 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 acurve.

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 pointi.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 Rules1. Derivative of a constant




2. Power Rule for derivatives





3. Derivative of a constant multiple






4. Sum and difference rules







5. Higher order derivatives







5. Higher order derivatives6. Product rule







5. Higher order derivatives6. Product rule

7. Quotient rule








6. Product rule

7. Quotient rule

8. Negative integer power rule





3. Derivative of a constant multiple4. Sum and difference rules


6. Product rule7. 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 functionsY= sin x




Y= cos x




Y= cos x

Y= tan x




Y= cos x

Y= tan x

Y= csc x




Y= cos x

Y= tan x

Y= csc x

Y= sec x




Y= cos x

Y= tan x

Y= csc x

Y= sec xY= 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 acurve.

i.e. average rate of change

Calculus derivatives

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