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DEFINITION

Calculus Derivatives

Calculus Derivatives

• If y = f(x), then the derivative of yor f(x)

with respect to x is defined as:

• Also denoted as

dy

dx

fx h fx

h

fx x fx

x

hx

=+− =+−

′

→→

lim ()()

lim ()()

,

00∆

∆

∆

ydf

dx fx Dy

h

x

′′

→

,,()

0

or

GENERAL RULES

d

dx cd

dx cx c

d

dx cx ncx d

dx unudu

dx

d

dx cu c du

dx

du

dx dx du

dy

dx

dy du

dx du

dy

dx

dy

du

du

dx

d

dx uvw du

dx

dv

dx

dw

dx

d

dx uv u dv

dx vdu

dx

d

dx uvw uv dw

dx uw dv

dx vw du

dx

d

dx

u

v

nn nn

==

==

==

=

=

±± ±

…=±± ±

…

=+

=++

−−

0

1

11

(chain rule)

(product rule)

()

=−v du dx u dv dx

v

()()

2(quotient rule)

EXPONENTIAL/LOGARITHMIC

TRIGONOMETRIC FUNCTIONS

d

dx uu

du

dx

sin cos

c

=

d

dx uu

du

dx

cos sin

t

=−

=

d

dx uu

du

dx

tan sec

c

=2

d

dx uu

du

dx

cot csc

s

=−

=

2

d

dx uuu

du

dx

sec sec tan

c

=

d

dx uuu

du

dx

csc csc cot

s

=−

=

d

dx u

u

du

dx u

d

dx u

u

du

dx

sin / sin /

cos cos

=

=

−

−< <

=−

−

<

−−

−

1

2

1

1

2

1

1

22

1

1

0

ππ

−−

−−

−−

−−

−

<

=+−< <

=−

+<<

=±

−

+< <

−< <

1

1

2

1

1

2

1

1

2

1

1

1

1

22

1

1

0

1

1

02

2

u

d

dx u

u

du

dx u

d

dx u

u

du

dx u

d

dx u

uu

du

dx

u

u

π

ππ

π

π

ππ

tan / tan /

cot cot

sec sec /

/sec

if

if

=±

−

+− < <

−< <

−−

−

d

dx u

uu

du

dx

u

u

csc /csc

csc /

1

2

1

1

1

1

20

02

if

if

π

π

HIGHER DERIVATIVES

DEFINITION

2nd Derivative

3

=

==

′′ =′′

=

d

dx

dy

dx

dy

dx

fx y

d

2

2()

(

3rd Derivative

n

=

=

d

dx

dy

dx

dy

dx

f

2

2

3

3

=′′′ =′′′

=

fx y

d

()

(

nth Derivative

=

=

== =

−

−

d

dx

dy

dx

dy

dx

fxy

n

n

n

n

nn

2

1

1()

() ()

LIEBNITZ’S RULE: HIGHER PRODUCT DERIVATIVES

Duv uDv DuD v DuD v vDu

d

nn

nnnnn

nn

() ()( ) ( )( ) ,

,,

=+

+

+…+

…

−−

1

1

2

22

12

2

where are the binomial coefficients

d

• Dp= operator dp/dxp,

so Dpu = dpu/dxp=

p

th

derivative of u

EXAMPLES

HIGHER DERIVATIVES

DEFINITION

2nd Derivative

3

=

==

′′ =′′

=

d

dx

dy

dx

dy

dx

fx y

d

2

2()

(

d

dx

uv u dv

dx

du

dx

dv

dx

vdu

dx

=+ +

2

2

2

2

2

2

2

d

dx

uv u dv

=

3

3

3

dx

du

dx

dv

dx

du

dx

dv

dx

vdu

dx

3

2

2

2

2

3

3

33+++

3rd Derivative

n

=

=

d

dx

dy

dx

dy

dx

f

2

2

3

3

=′′′ =′′′

=

fx y

d

()

(

nth Derivative

=

=

== =

−

−

d

dx

dy

dx

dy

dx

fxy

n

n

n

n

nn

2

1

1()

() ()

LIEBNITZ’S RULE: HIGHER PRODUCT DERIVATIVES

Duv uDv DuD v DuD v vDu

d

nn

nnnnn

nn

() ()( ) ( )( ) ,

,,

=+

+

+…+

…

−−

1

1

2

22

12

2

where are the binomial coefficients

d

• Dp= operator dp/dxp,

so Dpu = dpu/dxp=

p

th

derivative of u

EXAMPLES

IMPLICIT DIFFERENTIATION

METHOD TIPS

d

dx ue

u

du

dx a

d

dx ud

dx uu

du

dx

d

dx aaa

du

dx

aa

e

uu

u

log log ,,

ln log

ln

=≠

==

=

01

1

d

dx ee

du

dx

uu

v

=

d

dx ud

dx ee

d

dx vu vu d

vvuvu v v

l

(ln) ln

ln ln

=

== =+

−

vu du

dx uu

dv

dx

vv

l

ln+

−1

L’HÔPITAL’S RULE

• If differentiable functions f(x) and g(x) vanish

at x= a so that f(a)/g(a) = 0/0, then

provided that the limit on the right either

exists or is ±

Differentiate both sides

of the equation with

respect to x

Remember to use the

chain rule on y as it is a

function of x

Solve for y'

Use y = eln[g(x) ] • Differentiate using the chain

rule, product rule, and properties of logarithms

as necessary

Take the logarithm of each and differentiate

implicitly using the properties of logarithms as

necessary • Solve for y'

Notes: An inverse function

is denoted as f–1; do not

mistake the superscript (–1)

for an exponent • For

example, sin–1 uπ 1/sin u,

sin–1 (sin u) = u(–p/2< u <

p/2), and sin(sin –1u) =

u(–1< u <1) • Similar relations

exist for other trigonometric

functions

lim ()

() lim ()

()

xa xa

fx

gx

fx

gx

→→

=′

′

∞

_

__

_

h(x)

1CALCULUS DERIVATIVES •A-797-8

l e a r n • r e f e r e n c e • r e v i e w

TM

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