Derivative2

Derivative
Trigonometric Functions Hyperbolic Trigonometric Function
u u
dx
y = sin(u) ? dy = cos( ). ? u u
dx
y = sinh(u) ? dy = cosh( ). ?
u u
dx
y = cos(u) ? dy = ?sin( ). ? u u
dx
y = cosh(u) ? dy = sinh( ). ?
u u
dx
y = tan(u) ? dy = sec2 ( ). ? h u u
dx
y = tanh(u) ? dy = sec 2 ( ). ?
u u
dx
y = cot(u) ? dy = ?csc2 ( ). ? h u u
dx
y = coth(u) ? dy = ?csc 2 ( ). ?
u u u
dx
y = sec(u) ? dy = sec( ) tan( ). ? h u u u
dx
y = sec h(u) ? dy = ?sec ( ) tanh( ). ?
u u u
dx
y = csc(u) ? dy = ?csc( )cot( ). ? h u u u
dx
y = csc h(u) ? dy = ?csc ( ) coth( ). ?
1- sin2 ( 2 ) 2sin(x2 ).cos(x2 ).(2x)
dx
y = x ? dy =
2- = sec2 (3 +1)? = 2sec(3x +1).sec(3x +1) tan(3x +1) .3
dx
y x dy
3-
sec ( )
sec( ).sec ( ) tan( ).sec( ).tan( )
sec( )
tan( )
2
2
x
x x x x x
dx
dy
x
y x ?
= ? =
4- tan(3 ) sec2 (3x).3 3sec2 (3x)
dx
y = x ? dy = =
Derivative
Natural Logarithms ln(x) Exponential Function ex
If u(x) is differential function of (x) and
( )
ln[ ( )]
u x
dx
du
dx
y = u x ? dy =
1- 2
ln( 2 ) 2
x
x
dx
y = x ? dy =
2-
3 7
ln( 3 7) 2 3 2
2
+ ?
+
= + ? ? =
x x
x
dx
y x x dy
3-
sin ( )
ln[sin ( )] 2sin( ) cos( ) 2
2
x
x x
dx
y = x ? dy =
If u(x) is differential function of (x) and
dx
e du
dx
y = eu(x) ? dy = u(x) .
1- (2 )
2 2
e x
dx
y = ex ? dy = x
2- sin( ) e sin( )[x cos(x) sin(x)]
dx
y = ex x ? dy = x x +
3- tan( ) etan( )[sec2 (x)]
dx
y = e x ? dy = x
Properties Of Natural Logarithms Properties Of Exponential Function
ln(x. y ) =ln(x) + ln(y)
ln( ) ln(x) ln(y)
y
x = ?
ex+ y = ex .e y
y
x
x y
e
e ? = e
ln(xr ) = r ln(x) ln(1) = 0 eln(x) = x (ex )r = erx