Quick Review on Calculus
Definition of a derivative
If y = f(x), the derivative of y or f(x) with respect to x is defined as
dy
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
𝑓(𝑥 + ∆𝑥) − 𝑓(𝑥)
= lim
= lim
∆x→0
dx h→0
ℎ
∆𝑥
where h = ∆x. The derivative represents the slope of the curve (i.e. y against x) at
the point (x, y). It is also denoted by y′, df/dx, or f′(x). The process of taking a
derivative is called differentiation.
General rules of differentiation
Let u, v, w, and y be functions of x; and a, b, c, and n are constants. We have
the following.
d
dx
(𝑐 ) = 0
d
dw
du
dv
(𝑢𝑢𝑢) = 𝑢𝑢
+ 𝑣𝑣
+ 𝑤𝑤
dx
dx
dx
dx
d
(𝑐𝑐) = 𝑐
dx
d 𝑢
𝑣(𝑑𝑑/𝑑𝑑) − 𝑢(𝑑𝑑/𝑑𝑑)
� �=
dx 𝑣
𝑣2
d
du dv dw
(𝑢 ± 𝑣 ± 𝑤 ± ⋯ ) =
± ±
±⋯
dx
dx dx dx
du
1
=
dx 𝑑𝑑/𝑑𝑑
d
dv
du
(𝑢𝑢) = 𝑢 + 𝑣
dx
dx
dx
dy 𝑑𝑑/𝑑𝑑
=
dx 𝑑𝑑/𝑑𝑑
d
(𝑐𝑥 𝑛 ) = 𝑛𝑛𝑥 𝑛−1
dx
d
du
(𝑐𝑐) = 𝑐
dx
dx
d 𝑛
𝑑𝑑
(𝑢 ) = 𝑛𝑢𝑛−1
dx
𝑑𝑑
dy dy du
=
dx du dx
(𝐶ℎ𝑎𝑎𝑎 𝑟𝑟𝑟𝑟)
Derivatives of trigonometric functions
d
du
sin 𝑢 = cos 𝑢
dx
dx
d
du
cot 𝑢 = −csc 2 𝑢
dx
dx
d
du
tan 𝑢 = sec 2 𝑢
dx
dx
d
du
csc 𝑢 = −csc 𝑢 cot 𝑢
dx
dx
d
du
cos 𝑢 = −sin 𝑢
dx
dx
d
du
sec 𝑢 = sec 𝑢 tan 𝑢
dx
dx
1
Derivatives of exponential and logarithmic functions
d
1 du
ln 𝑢 =
dx
𝑢 dx
d 𝑢
du
𝑎 = 𝑎𝑢 ln 𝑎
dx
dx
d 𝑢
du
𝑒 = 𝑒𝑢
dx
dx
d
log 𝑎 𝑒 du
log 𝑎 𝑢 =
dx
𝑢 dx
Higher derivatives
2
d 𝑑𝑑
Second derivative = 𝑑𝑑𝑥𝑦2 = dx
� � = 𝑓 ′′ (𝑥) = 𝑦′′
𝑑𝑑
𝑑𝑛 𝑦
d
𝑑𝑛−1 𝑦
nth derivative = 𝑑𝑥 𝑛 = dx �𝑑𝑥 𝑛−1 � = 𝑓 (𝑛) (𝑥) = 𝑦 (𝑛)
Increasing and decreasing functions
dy
For strictly increasing function: dx > 0
dy
For strictly decreasing function: dx < 0
Stationary points of a curve
If y = f(x), then the maximum and minimum points in y are regarded as stationary
dy
points, where dx = 0.
𝑑2 𝑦
For maximum point: 𝑑𝑥 2 < 0.
𝑑2 𝑦
For minimum point: 𝑑𝑥 2 > 0.
2
Definition of an indefinite integral
dy
If dx = 𝑓(𝑥), then 𝑦 is the function whose derivative is 𝑓(𝑥) and is called the
anti-derivative of 𝑓(𝑥) or the indefinite integral of 𝑓(𝑥), denoted by ∫ 𝑓(𝑥) 𝑑𝑑.
That is to say, y = ∫ 𝑓(𝑥) 𝑑𝑑. Since the derivative of a constant is zero, all indefinite
integrals differ by an arbitrary constant.
General formulae of integrals
Let u, v, and w be functions of x; and a, b, c, and n are constants. We have the
following.
� 𝑎 𝑑𝑑 = 𝑎𝑎
� 𝑎𝑎(𝑥) 𝑑𝑑 = 𝑎 � 𝑓(𝑥) 𝑑𝑑
�(𝑢 ± 𝑣 ± 𝑤 ± ⋯ ) 𝑑𝑑 = � 𝑢 𝑑𝑑 ± � 𝑣 𝑑𝑑 ± � 𝑤 𝑑𝑑 ± ⋯
𝑢𝑛+1
∫ 𝑢𝑛 𝑑𝑑 = 𝑛+1
(n ≠ −1)
� 𝑒 𝑢 𝑑𝑑 = 𝑒 𝑢
(Integration by parts)
𝑑𝑑
∫ 𝑢 = ln 𝑢 if u > 0 or ln(−𝑢) if u < 0
∫ 𝑢 𝑑𝑑 = 𝑢𝑢 − ∫ 𝑣 𝑑𝑑
Integrals of trigonometric functions
� sin 𝑢 𝑑𝑑 = − cos 𝑢
� tan 𝑢 𝑑𝑑 = ln sec 𝑢 = − ln cos 𝑢
� cos 𝑢 𝑑𝑑 = sin 𝑢
� cot 𝑢 𝑑𝑑 = ln sin 𝑢
� sec 2 𝑢 𝑑𝑑 = tan 𝑢
� csc 2 𝑢 𝑑𝑑 = −cot 𝑢
𝑢 𝜋
� sec 𝑢 𝑑𝑑 = ln(sec 𝑢 + tan 𝑢) = ln tan � + �
2 4
� csc 𝑢 𝑑𝑑 = ln(csc 𝑢 − cot 𝑢) = ln tan
3
𝑢
2
Definition of a definite integral
Let f(x) be defined in an interval a ≤ x ≤ b. Divide the interval into n equal parts
of length ∆x = (b − a)/n. Then the definite integral of f(x) between x = a and
x = b is defined as
𝑏
� 𝑓(𝑥) 𝑑𝑑 = lim {𝑓(𝑎) ∆𝑥 + 𝑓(𝑎 + ∆𝑥) ∆𝑥 + 𝑓(𝑎 + 2∆𝑥) ∆𝑥 + ⋯ + 𝑓(𝑎 + (𝑛 − 1)∆𝑥) ∆𝑥}
𝑛→∞
𝑎
The limit will certainly exist if f(x) is piecewise continuous.
d
If f(x) = dx 𝑔(𝑥), then by the fundamental theorem of the integral calculus the above
definite integral becomes
𝑏
𝑏
𝑑
𝑏
𝑔(𝑥)𝑑𝑑 = � 𝑑{𝑔(𝑥)} = 𝑔(𝑥) � = 𝑔(𝑏) − 𝑔(𝑎)
𝑎
𝑎 𝑑𝑑
𝑎
� 𝑓(𝑥) 𝑑𝑑 = �
𝑎
𝑏
Simple rules of definite integrals
Let f, g, and h be functions of x; and a, b, and c are constants. We have the
following.
𝑎
𝑏
� 𝑓(𝑥) 𝑑𝑑 = 0
𝑎
𝑎
� 𝑓(𝑥) 𝑑𝑑 = − � 𝑓(𝑥)𝑑𝑑
𝑎
𝑏
𝑎
𝑎
𝑏
𝑏
𝑏
𝑏
� 𝑐 𝑓(𝑥) 𝑑𝑑 = 𝑐 � 𝑓(𝑥)𝑑𝑑
𝑐
𝑏
� 𝑓(𝑥) 𝑑𝑑 = � 𝑓(𝑥)𝑑𝑑 + � 𝑓(𝑥)𝑑𝑑
𝑎
𝑎
𝑏
𝑐
𝑏
𝑏
� {𝑓(𝑥) ± 𝑔(𝑥) ± ℎ(𝑥) ± ⋯ } 𝑑𝑑 = � 𝑓(𝑥)𝑑𝑑 ± � 𝑔(𝑥)𝑑𝑑 � ℎ(𝑥)𝑑𝑑 ± ⋯
𝑎
𝑎
4
𝑎
𝑎