Calculus 2.1: Differentiation Formulas

d

dxx nxn n( ) 1

d

dxc( ) 0A. Derivative of a

Constant:

B. The Power Rule:

C. Constant Multiple Rule:

d

dxcf x c

d

dxf x( ) ( )

D. Sum Rule:d

dxf x g x

d

dxf x

d

dxg x( ) ( ) ( ) ( )

E. Difference Rule:

d

dxf x g x

d

dxf x

d

dxg x( ) ( ) ( ) ( )

d

dxf x g x f x

d

dxg x g x

d

dxf x[ ( ) ( )] ( ) ( ) ( ) ( )

( )fg f g g f

a f x x x) ( ) ( )( ) 6 73 4

F. Product Rule:

Ex 1: Find f ‘(x):

b y x x) ( )( ) 2 5 1

G. Quotient Rule

fg

gf fg

g

2

( ) ( ) ( ) ( )lo d hi hi d lo

lolo

Ex 2: Find y’

a yx x

x)

2

3

2

6b y

x

x)

1 2

d

dx

g x f x f x g x

g xf xg x

ddx

ddx( )

( )

( ) ( ) ( ) ( )

[ ( )]

2

A. Higher Derivatives

1. Second Derivative: f’’ = (f’)’

(if f’ is differentiable) yd

dx

dy

dx

d y

dx

2

2

2. Third Derivative: f’’’ = (f’’)’(if f’’ is differentiable)

yd

dx

d y

dx

d y

dx

2

2

3

3

Note:y

d

dxy

d y

dxn n

n

n( ) ( ) 1 “y super n”

means the nth derivative

Calculus 2.2: Differentiation Problems

B. Examples

1. Find the equation of a tangent line at the point (1, ½) to the curve:

2. Find the points on the curve y = x4 – 6x2 + 4 where the tangent line is horizontal

3. At what points on the hyperbola xy = 12 is the tangent line parallel to 3x + y = 0?

4. If h(x) = xg(x) and it is known that g(3) = 5 and g‘(3) = 2, find h‘(3)

5. #37 p.124

y x x ( )1 2

Calculus 2.3: More Rates of Change

A. Average Rate of Change of y with respect to x:

y

x

f x f x

x x

( ) ( )2 1

2 1

B. Instantaneous Rate of Change: (derivative)

dy

dx

y

xx

lim

0

C. Applications

1. Linear Motion:a) velocity – the derivative

of the position function s = f(t)

b) speed – the absolute value of velocity

c) acceleration – the derivative of the velocity with respect to time

d) jerk – the derivative of acceleration

v tds

dt( )

speed v t ( )

a tdv

dt

d s

dt( )

2

2

j tda

dt

d s

dt( )

3

3

2. Economics:

a. marginal cost – the rate of change of cost with respect to level of production

b. marginal revenue – the derivative of the revenue function

dc

dx

dr

dx

Calculus 2.4: Derivatives of Trigonometric Functions

d

dxx x(sin ) cos

d

dxx x(cos ) sin

d

dxx x(tan ) sec 2

A. Derivatives of Trig Functions: (Memorize!!)

d

dxx x(cot ) csc 2

d

dxx x x(csc ) csc cotd

dxx x x(sec ) sec tan

Graphing Sin/Cos Functions

Graphing Cos/-Sin Functions

Calculus 2.5: The Chain Rule

dy

dx

dy

du

du

dx

A. The Chain Rule:

If f and g are both differentiable and F = f o g, then F is differentiable and F’ = f ‘(g(x))g‘(x)

Ex 1:

F x x( ) 2 1

B. Power Rule Combined with Chain Rule

d

dxg x n g x g x

n n( ) ( ) ( ) 1

or

d

dxu nu

du

dxn n 1

C. The Chain Rule (Trigonometric Functions)

1 2. siny x

2 2. siny x3. ( ) sin(cos(tan ))f x x

4. secy x

5 5. siny x at x 3

Find tangent line

Calculus 2.6: Implicit Differentiation

dy

dx

1 12 2.x y

2 63 3.x y xy

A. Method of Implicit Differentiation:

1. Differentiate both sides of the equation with respect to x

2. Solve the resulting equation for

B. Examples—find y’ 3

6 422

.xy

x y

4 12. y x

B. Finding

1. Use implicit differentiation to find

2. Differentiate

3. Substitute the expression for

4. Simplify

dy

dx

d y

dx

2

2

d y

dx

2

2

dy

dxdy

dx

C. Implicit Differentiation with Trigonometric Functions

1. sin(x+y) = y2 cos x2. tan(x/y) = x + y3. 4 cos x sin y = 14. 2y = x2 + sin y

Calculus Unit 2 Test

Grademaster #1-30 (Name, Date, Subject, Period, Test Copy #)

Do Not Write on Test! Show All Work on Scratch Paper!

Label BONUS QUESTIONS Clearly on Notebook Paper. (If you have time)

Find Something QUIET To Do When Finished!