TABLE OF DERIVATIVES, DEFINITIONS AND IDENTITIES

Chain Rule Product Rule Quotient Rule

n
Dx(u )= nu n-1
Dx u Dx(uv)= u Dx v + v Dx u Dx ( uv ) =
v D x uu D x v
v2

Trigonometric Function -1
1
Dx(cos u)= 1u 2 Dx u
Dx (sin u) = cos u Dx u 1
-1
Dx(tan u)= 1+u
2
Dxu
Dx (cos u) = sin u Dx u
1
-1
Dx (tan u) = sec u Dx u 2
Dx(csc u)= u u21 Dx u
1
Dx (csc u) = csc u cot u Dx u -1
Dx(sec u)= u u21 Dx u
Dx (sec u) = sec u tan u Dx u 1
-1
Dx(cot u)= 1+u
2
Dx u
Dx (cot u) = csc2u Dx u

Hyperbolic Function Inverse Hyperbolic Function

1
-1
Dx(sinh u)= cosh u Dx u Dx(sinh u) = u2 +1 Dx u
1
Dx(cosh u)=sinh u Dx u -1
Dx(cosh u) = u2 1 Dx u
2
Dx(tanh u)=sech u Dx u 1
-1
Dx(tanh u) = 1u2 Dx u
Dx(csch u)= csch u coth u Dx u
1
-1
Dx(sech u)= sech u tanh u Dx u Dx(csch u) = u 1+u2 Dx u
1
Dx(coth u)= csch2u Dx u -1
Dx(sech u) = u 1u 2 Dx u
Inverse Trigonometric Function 1
-1
1 Dx(coth u) = 1u2 Dx u
Dx(sin-1 u)= 1u 2 Dx u

Exponential and Logarithmic Function

 Dx(eu)= eu Dx u
Dx(au)= au In a Dx u
1
Dx(In u)= u Dx u
1
Dx(loga u)= u logae Dx u

Trigonometric Identities 1
cos = sin2
1 sin 2

sin = csc
sin (2 ) = 2 sin cos
1
cos
= sec cos (2 = cos2 sin2

sin 1 = 2 cos2 q 1
= 1 2 sin2 q
tan = cos = cot
1+cos
2

cos 1 cos 2 = 2
cot = sin = tan
1cos
sin 2
cos+
=1 2 sin2 2 = 2

1+tan2 = sec2
=sin
cos ( 2
1+ cot2 = csc2

cos( + 2 ) = cos sin( ) = sin
+
sin( + 2 ) = sin sin( ) = sin

tan( + ) = tan sin ( 2
=cos

cos( ) = cos
cos( ) = cos
sin( ) = sin
cos( + ) = cos
tan( = tan Hyperbolic Definition & Identities
cos(x + y) = cos x cos y sin x sin y x
e e
x

cos(x y) = cos x cos y + sin x sin y sinh = 2
sin(x + y) = sin x cos y + cos x sin y x
e +e
x

sin(x y) = sin x cos y cos x sin y cosh = 2
tan x+ tan y
tan(x+y) = 1tan x tan y e e
= x x
x x

tanh e +e
tan xtan y
tan(x y) = 1+ tan x tan y 2

csch = x
e ex
 2
1
tanh2 = sech2
sech = e +e
x
x

x x
1 coth2 = csch2
e +e
=
coth x
e e
x

cosh(x + y) = cosh x cosh y + sinh x sinh y

+ sinh(x + y) = sinh x cosh y + cosh x sinh y
cosh sinh = ex

cosh sinh = e-x cos h 1

1
sinh-1=In(x+ x +1 )
2
sinh2 2 = 2
)
cosh-1=In(x+ x 1 )
2

cos h +1

1 1+ x 1
tanh-1=
2 1 x
cosh2 2 = 2
)

1+ 1+ x 2
-1
csch =In x
sinh (2 ) = 2 sinh cosh
1+ 1x 2
-1 cosh (2 = cosh2 + sinh2
sech =In x
1 x +1 John Joshua F. Montaez
-1
coth = 2 x1
GE21||Differential Calculus
September 12, 2011

cosh2 sinh2 =1