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Reference Guide

Calculus Derivatives - Reference Guides

4 pages2269 viewsFall 2015

Department
BAD - Business Administration
Course Code
BAD 200
Professor
All
Chapter
Permachart

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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()
(
=′′′ =′′′
=
fx y
d
()
(
nth Derivative
=
=
== =
d
dx
dy
dx
dy
dx
fxy
n
n
n
n
nn
2
1
1()
() ()
LIEBNITZS 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+++
=′′′ =′′′
=
fx y
d
()
(
nth Derivative
=
=
== =
d
dx
dy
dx
dy
dx
fxy
n
n
n
n
nn
2
1
1()
() ()
LIEBNITZS 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
permacharts
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www.permacharts.com
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