Appendix A: Mathematics
153
A.4 Derivatives
•
d
dX
1
X
= −
1
X 2
•
d
dX
X
n
= n × X
n−1
•
d
dX
n
√
X =
1
n ×
n
√
X n−1
•
d
dX
sin(X ) = cos(X )
•
d
dX
cos(X ) = − sin(X )
•
d
dX
ln(X ) =
1
X
•
d
dX
|X | =
−1 for X < 0
1 for X > 0
•
d
dX
( f (X ) × g(X )) =
d f (X )
dX
g(X ) + f (X )
dg(X )
dX
(product rule)
•
d
dX
f (X )
g(X )
=
d f (X )/dX × g(X ) − f (X ) × dg(X )/dX
[g(X )]
2
(quotient rule)
A.5 Integrals
The indefinite integral or antiderivative F(X ) =
f (X )dX + c of a function f (X ) is
a differentiable function F(X ) whose derivative is equal to f (X ), i.e.,
dF(X )
dX
= f (X ).
The definite integral of a continuous real-valued function f (X ) on a closed interval
[a, b], i.e.,
b
a f (d)dX = F(b) − F(a), is represented by the area under the curve
f (X ) from X = a to X = b.
Some selected antiderivatives (c: arbitrary constant of integration):
•
e
X dX = e
X
+ c
•
√
X dX =
2
3
X
3
2 + c
•
sin(X )dX = − cos(X ) + c
•
cos(X )dX = sin(X ) + c
•
sin(αX ) · cos(αX )dX =
1
2α
sin
2
(αX ) + c
•
sin
2
(αX )dX =
1
2
(X − sin(αX ) cos(αX )) + c =
1
2
(X −
1
2α
sin(2αX )) + c
•
cos
2
(αX )dX =
1
2
(X + sin(αX ) cos(αX )) + c =
1
2
(X +
1
2α
sin(2αX )) + c