Sheng-Fang Huang

Definition of DerivativeThe Basic Concept

Definition of DerivativeDefinition

Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit:

provided this limit exists. The derivative of f is also written as f’, or df/dx.

If this limit exists for each x in an open interval I (e.q. from -∞ to + ∞), then we say that f is differentiable on I.

x

yh

xfhxfx

xfxxf

dx

df

h

0x

0

0x

lim

lim

lim

)()(

)()(

Standard Derivative

rule)(Chain

''

'')'(

'')'(

)constant ( ')'(

2

'

dx

dy

dy

du

dx

du

v

uvvu

v

u

uvvuuv

vuvu

ccucu

Standard Derivative y=f(x)

dx

df

1 nx 1nnx 2 xe xe 3 kxe kxke 4 xa aa x ln 5 x ln

x

1

6 xalog ax ln

1

7 sin x cos x 8 cos x - sin x 9 tan x sec2 x

10 cot x -cosec2 x 11 sec x sec x tan x 12 cosec x -cosec x cot x 13

arcsin x (sin-1x) 21

1

x

14 arccos x

21

1

x

15 arctan x

21

1

x

16 arccot x 21

1

x

17 sinh x ( (ex+e-x)/2 ) cosh x 18 cosh x ( (ex-e-x)/2 ) sinh x

Exercises

1. y = xx tan5 4. y = xxx sincos4

2. y = xe x ln3 5. y = 5a (a is constant)

3. y =x

x

2

log5

Compute dy/dx

Functions of a function (1)sinx is a function of x since the value of sinx

depends on the value of the angle x. Simiarly, sin(2x+5) is a function depends on

________. Since (2x+5) is also a function of x, we say

sin(2x+5) is a function of a function of x.By Chain Rule:

Let y = sin(2x+5), u = 2x+5.

)52cos(22cos

xudx

dydx

du

du

dy

dx

dy

Functions of a function (2)Products

Compute the derivative of e2xln5x.

QuotientsCompute the derivative of .2

2cos

x

x

Logarithmic DifferentiationThe rule for differentiating a product or a quotient is used

when there are only two factors, i.e. uv or u/v. Where there are more than two functions, the derivative is best found by ‘logarithmic differentiation’.

Let , where u, v, and w are functions of x.Take logs to the base e on both sides:

Rule)(Chain 1

)(ln

1)(ln

dx

dF

FF

dx

dx

xdx

d

w

uvy

dx

dw

wdx

dv

vdx

du

uy

dx

dy

dx

dw

wdx

dv

vdx

du

udx

dy

y

wvuy

111

1111

lnlnlnln

ExercisesSolve the derivative of .

y = x4e3xtanx.

x

xxy

2cos

sin2

Implicit FunctionsExplicit function:

If y is completely defined in terms of x, y is called an explicit.

E.g. y = x2 – 4x + 2Implicit function

E.g. x2 + y2 = 25, or x2 + 2xy + 3y2 = 4.

The differentiation of an implicit function:

yxdx

dyyx

dx

dyyy

dx

dyxx

y

x

dx

dy

dx

dyyx

22)62( 06)22(2

. 022

Partial DifferentiationThe volume V of a cylinder of radius r and

height h is given by

V depends on two quantities, r and h. Keep r constant, V increases as h increases. Keep h constant, V increases as r increases.

is called the partial derivative of V with respect to h where r is constant.

Let z = 3x2 + 4xy – 5y2.Compute and .

hrV 2

h

V

dh

dV

r

constant

x

z

y

z

Standard Integrals (1)1

1,1

1

nCn

xdxx

nn

2 Cedxe xx

3 C

k

edxe

kxkx

4 C

a

adxa

xx ln

5 Cxdx ln

x

1

6 Cxxdx cossin

7 Cxxdx sincos

8 Cxxdx coslntan

9 Cxxdx sinlncot

10 Cxxxdx tanseclnsec

11 Cxxxdx cotcsclncsc

12 C

a

x

adx

ax

arctan11

22

13

dxax 22

1_______________

14

dxax 22

1_______________

15 Cxxxxdx lnln

Function of a linear function of xWe know that , but how about

?

Solution: Let z = 5x-4. Change the original equation into the

following form:

We have

Now, try to solve and .

Cx

dxx7

76

dxx 6)45(

dzdz

dxzdxz 66

5

15)45(

dz

dxx

dx

d

dx

dz

C

xC

zdzzdz

dz

dxzdxz

35

)45(

75

1

5

1 77666

dxe x 43 dxx )52cos(

and Formula:

How to prove?

Exercises:

dxxfxf )()( ' dxxf

xf

)(

)('

Cxfdxxf

xf))(ln(

)(

)('

Cxf

dxxfxf 2

))(()()(

2'

dxxx

x

53

322

xdxtan

Integration of productsHow to integrate x2lnx?Integration by parts:

Solve .

vduuvudv

dxdx

duvuvdx

dx

dvu

dxdx

duvdx

dx

dvuuv

dx

duv

dx

dvuuv

dx

d)(

xdxx ln2

Integration by partial fractionsExample:

Rules:degree of numerator < degree of denominator

If not, first of all, divide out by the denominator.Denominator must be factorized into prime factors

(important!). (ax+b) gives partial fractions A/(ax+b) (ax+b)2 gives partial fractions A/(ax+b) + B/(ax+b)2

(ax+b)3 gives partial fractions A/(ax+b) + B/(ax+b)2 + C/(ax+b)3

ax2+bx+c gives partial fractions (Ax+B)/(ax2+bx+c)

Cxxdxx

dxx

dxxx

dxxx

xdx

xx

x

)1ln(2)2ln(31

2

2

3

)1

2

2

3(

)2)(1(

1

23

12

Integration by partial fractionsExample:

)1()1)(1()1(

)1)(1(by sidesboth Multiply )1(11)1)(1(

)1)(1(1

22

222

2

2

2

23

2

xCxxBxAx

xxx

C

x

B

x

A

xx

x

dxxx

xdx

xxx

x

Integration of Trigonometrical FunctionsBasic trigonometrical formula:

)cos()cos(sinsin2

)cos()cos(coscos2

)sin()sin(sincos2

)sin()sin(cossin2

)2cos1(2

1cos

)2cos1(2

1sin

1cossin

2

2

22

BABABA

BABABA

BABABA

BABABA

xx

xx

xx

Integration of Trigonometrical Functions

xdxxdxdxxxxxxdxx

dxx

xdx

xdxxxdxxxdx

dxxxdx

Cxdx

Cxxdx

2sin6sin2

1))24sin()24(sin(

2

12cos4sin

)2

2cos1(sin

sin)2cos1(2

1sinsinsin

)2cos1(2

1sin

sincos

cossin

24

23

2

Multiple IntegrationDouble integral

A double integral can be evaluated from the inside outward.

Key point: when integrating with respect to x, we consider y as constant.

2

1

2

1

),(y

y

x

xdxdyyxf

2

1

2

1

),( yy

yy

xx

xxdydxyxf

12

Multiple IntegrationA double integral can also be

sometimes expressed as:

2

1

2

1

),(xx

xx

yy

yydydxyxf

z = f(x,y)

x

y

dxdy

Multiple IntegrationExample

If . Compute V. 6

0

4

0

22 )( dydxyxV