AP Calculus BC

Chapter 3

3.1 Derivative of a FunctionDefinition of the Derivative: 0

( ) ( )lim '( )h

f x h f x f xh

Notation:''( )yf xdydxdfdx

If f’ exists we say f has a derivative and is differentiable at x.

Can have one-sided derivatives and two-sided derivatives.

Graph Examples

Alternative forms are discussed in the HW.

Examples:2 32 105

512

y x

y x x

y x x

3.2 DifferentiabilityWhere a f’ might fail to exist 1. Corner 2. Cusp

3. Vertical Tangent 4. Discontinuity

Differentiability implies local linearity.EXAMPLES:

( ) 2 3

2( )3

f x x

xg xx

Differentiability implies continuity.

Calculator: nderiv(f(x), x, a) = slope at aGraphing: 2 1( , , )Y nderiv Y x x

EXAMPLES:on calculator

2

3

( )( )( ) ln

f x xf x xf x x

Intermediate Value Theorem for Derivatives

3.3 Rules for Differentiation

Derivative of a constant:Power Rule:

Constant Multiple Rule:Sum/Diff. Rule:

u and v are functions of x( ) 0d c

dx

1( )n nd x nxdx

( ) 'd cu cudx

( ) ' 'd u v u vdx

Product Rule:( ) ' 'd uv u v uv

dx

Quotient Rule:

2' '( )d u vu uv

vdx v

The first derivative tells us the slope of the tangent line.

3.3 Examples4 3 2

3 2

23

5( ) 3 7 3 223

( ) ( 1)( 3)

3( )4

f x x x x x

f x x x

xf xx

Find the first derivative of the following problems.

Find the Horizontal Tangents for:

4 2( ) 2 2f x x x

3.4 Velocity & other Rates of Change

Instantaneous Rate of Change is at a specific time. (derivative)Rate of change for the Area of Circle:

2A r 2dA rdr

Evaluate whenr = 5, and r =10

PositionVelocity

Acceleration

( )'( ) ( )''( ) '( ) ( )

s ts t v ts t v t a t

Displacement - sAvg. Velocity -

st

Instantaneous velocity is s’ or v, plug in the value.Speed is absolute value of velocity.A particle is at rest when velocity = 0.

Derivatives in Economics – “Marginal” Cost, Revenue

3.5 Derivatives of Trig. FunctionsOn calculator, find the derivative of y = sin x

(sin )

(tan )

(sec )

d xdxd xdxd xdx

cos x

sec x tan x

2sec x

(cos )

(cot )

(csc )

d xdxd xdxd xdx

2csc x

- sin x

- csc x cot x

EXAMPLES:2

tan( )

( ) 4 coscos( )1 sin

xf x xf x x x

xf xx

Tangent and Normal lines do not change.Calculate everything the same…….

3.6 Chain RuleChain Rule is the “Outside – Inside Rule”

( ( ( )) '( ( )) '( )d f g x f g x g xdx

2 3(3 1)y x Example:2 23(3 1) (6 )dy x x

dx

Examples:

2

3

4 2

cos( 1)

sin ( )

tan (3 2)

4 (sec(3 ))

y x

y x

y x

y x x

T.A.

P.T.

P.T.A.

Product Rule with Chain Rule

Answers:22 sin( 1)y x x

23sin ( )cos( )y x x

3 2 2 224 tan (3 2)sec (3 2)y x x x

4sec(3 ) 12 sec(3 )tan(3 )y x x x x

3.6 cont’d.Slopes of Parametric Curves

( )

( )

x f t

y g t

dydy dtdx dx

dt

Example: Find the Tangent and Normal Linesec( )

tan( )

4

x t

y t

t

T.L.

N.L.

1 2( 2)

11 ( 2)2

y x

y x

3.7 Implicit Differentiation( )f xory

Now, non-functions……..

2y x First Derivative:2 ' 1

1'2

yy

yy

Examples:2 2

2

2 2

25

2 sin

7

x y

y x y

x xy y

Find T.L. & N.L. at (-1,2)

Find the second derivative of:

3 22 3 8x y

Power rule for Rational Exponents……

3.8 Derivatives of Inv. Trig.1sin

sin

y xory x

Triangle.. First derivative??? 2

1'1

yx

For the following problems: u = f(x)

12

12

12

1(sin )1

1(tan )1

1(sec )1

d duudx dxu

d duudx dxu

d duudx dxu u

1 1cos sin2

x x 1 1cot tan2

x x 1 1csc sec2

x x

12

12

12

1(cos )1

1(cot )1

1(csc )1

d duudx dxu

d duudx dxu

d duudx dxu u

Examples????

3.9 Deriv. of Exponential/LogarithmicLong Form Derivative of : xy e

01lim 1

h

heh

For the rules: u = f(x)

( )

( ) ln

u u

u u

d due edx dx

d dua a adx dx

1(ln )

1(log )lna

d duu udx dx

d duudx u a dx

1( )n nd duu nudx dx

Examples:(Be careful of #6)2

3

2 3

4 3

ln( 3)

ln 3

x x

x

x

x

y e

y x

y x

y x x x

y x e

y x

3y x

Do not forget Product & Quotient Rules…….