Comprehensive Formulas of Limits,
Differentiation, and Integration
1 Introduction
In this document, we present the essential formulas and concepts from three
critical areas of calculus: limits, differentiation, and integration. These concepts
form the foundation for more advanced topics in calculus and analysis. Each
section covers both fundamental rules and advanced methods.
2 Limits
The limit is a fundamental concept that describes the behavior of a function as
its input approaches a particular value.
2.1 Basic Limit Formulas
• Limit of a constant:
lim k = k
x→c
• Limit of xn :
lim xn = cn
x→c
• Sum and Difference Rule:
lim [f (x) ± g(x)] = lim f (x) ± lim g(x)
x→c x→c x→c
• Product Rule:
lim [f (x) · g(x)] = lim f (x) · lim g(x)
x→c x→c x→c
• Quotient Rule:
f (x) limx→c f (x)
lim = , g(c) ̸= 0
x→c g(x) limx→c g(x)
• Power Rule:
lim xn = cn
x→c
• Squeeze Theorem: If f (x) ≤ g(x) ≤ h(x) and limx→c f (x) = limx→c h(x) =
L, then limx→c g(x) = L.
3.1 Basic Differentiation Rules
• Constant Rule:
d
[c] = 0
dx
• Power Rule:
d n
[x ] = nxn−1
dx
• Sum/Difference Rule:
d d d
[f (x) ± g(x)] = [f (x)] ± [g(x)]
dx dx dx
• Product Rule:
d
[u(x) · v(x)] = u′ (x)v(x) + u(x)v ′ (x)
dx
• Quotient Rule:
v(x)u′ (x) − u(x)v ′ (x)
d u(x)
=
dx v(x) [v(x)]2
• Chain Rule:
d
[f (g(x))] = f ′ (g(x)) · g ′ (x)
dx
3.2 Derivatives of Common Functions
• Exponential Functions:
d x d x
[e ] = ex , [a ] = ax ln(a)
dx dx
• Logarithmic Functions:
d 1 d 1
[ln(x)] = , [loga (x)] =
dx x dx x ln(a)
• Trigonometric Functions:
d d
[sin(x)] = cos(x), [cos(x)] = − sin(x)
dx dx
d
[tan(x)] = sec2 (x)
dx
• Inverse Trigonometric Functions:
d 1 d 1
[sin−1 (x)] = √ , [tan−1 (x)] =
dx 1 − x2 dx 1 + x2
2
3.3 Higher-Order Derivatives
The second derivative and higher derivatives give information about the con-
cavity and the rate of change of the rate of change of a function:
d2 dn
f ′′ (x) = [f (x)], f (n) (x) = [f (x)]
dx2 dxn
4 Integration
Integration is the inverse process of differentiation and is used to find the total
accumulated quantity, such as area under a curve.
4.1 Basic Integration Rules
• Constant Rule: Z
k dx = kx + C
• Power Rule:
xn+1
Z
xn dx = +C (for n ̸= −1)
n+1
• Sum/Difference Rule:
Z Z Z
(f (x) ± g(x)) dx = f (x) dx ± g(x) dx
• Constant Multiple Rule:
Z Z
kf (x) dx = k f (x) dx
4.2 Common Integrals
• Exponential Functions:
Z
ex dx = ex + C
ax
Z
ax dx = +C
ln(a)
3
• Logarithmic Functions:
Z
ln(x) dx = x ln(x) − x + C
• Trigonometric Functions:
Z
sin(x) dx = − cos(x) + C
Z
cos(x) dx = sin(x) + C
Z
tan(x) dx = ln | sec(x)| + C
4.3 Integration Techniques
• Integration by Substitution:
Z Z
f (g(x))g ′ (x) dx = f (u) du
• Integration by Parts:
Z Z
u dv = uv − v du
• Trigonometric Substitution:
x = a sin(θ), x = a tan(θ), x = a sec(θ)
• Partial Fractions: Decompose rational functions into simpler fractions.
4.4 Definite Integrals
Z b
f (x) dx = F (b) − F (a)
a
4
4.5 Numerical Methods
• Trapezoidal Rule:
b
b−a
Z
f (x) dx ≈ (f (a) + f (b))
a 2
• Simpson’s Rule:
b
b−a
Z
a+b
f (x) dx ≈ f (a) + 4f + f (b)
a 6 2
5 Conclusion
This document covers the fundamental rules and advanced methods for limits,
differentiation, and integration. These concepts are essential for further study
in calculus and form the basis for much of mathematical analysis.
