Calculus is a branch of mathematics that deals with the study of rates of change and accumulation.

There are two main branches of calculus: differential calculus and integral calculus. Here are some common formulas and theorems used in each branch:

Differential:

1. Limit:
   • lim x->a f(x) = L
2. Derivative:
   • f'(x) = lim (h->0) (f(x+h) – f(x))/h
   • Power rule: f(x) = x^n , f'(x) = nx^(n-1)
   • Constant multiple rule: f(x) = cg(x), f'(x) = cg'(x)
   • Sum rule: f(x) = g(x) + h(x) , f'(x) = g'(x) + h'(x)
   • Chain rule: f(g(x)) = g'(x) * f'(g(x))

Integral:

1. Indefinite Integral:
   • ∫f(x)dx = F(x) + C
2. Definite Integral:
   • ∫^b_af(x)dx = F(b) – F(a)
3. Fundamental Theorem of Calculus:
   • ∫^b_af(x)dx = F(b) – F(a) = F(x) = f(x)dx = dF/dx
4. Integration by Substitution:
   • ∫f(g(x))g'(x)dx = ∫f(u)du
5. Integration by Parts:
   • ∫u(x)dv(x) = u(x)v(x) – ∫v(x)du(x)
6. Trigonometric Integrals:
   • ∫sin(x)dx = -cos(x) + C, ∫cos(x)dx = sin(x) + C
   • ∫sec^2(x)dx = tan(x) + C, ∫sec(x)tan(x)dx = sec(x) + C
   • ∫sec(x)dx = ln|sec(x) + tan(x)| + C

*Note: This is not an exhaustive list of all formulas, only some of the most common formulas were mentioned above. It is a very wide subject, and various other formulas and theorems are present for specific areas of study such as multivariable, vector etc.

Basics

It is a branch of mathematics that deals with rates of change and accumulated quantities. There are two main branches of calculus: differential calculus and integral calculus. Differential calculus concerns the study of the derivative of a function, which gives information about the instantaneous rate of change of the function at a particular point. Integral calculus concerns the study of the integral of a function, which gives information about the accumulated change of the function over an interval. These two branches are connected by the fundamental theorem of calculus.

Fundamental Theorem

The fundamental theorem of calculus is a fundamental result in mathematical analysis that connects the concept of the derivative of a function with the concept of the function’s integral. There are two parts to the theorem. The first part states that if F(x) is an antiderivative of f(x), then the definite integral of f(x) over the interval [a,b] is equal to F(b) – F(a). The second part states that if f(x) is a continuous function, then F(x) = ∫f(x) dx is an antiderivative of f(x), where ∫ denotes the integral symbol and dx denotes an infinitesimal element. These two parts connect the concepts of differentiation and integration and provide a link between the two branches of calculus.

It is used in a wide range of applications in various fields of science, engineering, and economics.

Some examples of its uses include:

1. Physics: It is used in physics to model the motion of objects, from baseballs to galaxies. It also plays a role in understanding waves, electricity, and magnetism.
2. Engineering: Engineers use it to optimize designs, such as finding the maximum strength of a bridge or the most efficient shape for an airplane wing.
3. Economics: used in economics to study how the economy behaves over time, such as how market prices change with supply and demand.
4. Medicine: In medicine, it is used in the study of how drugs are metabolized in the body and how they interact with biological systems.
5. Computer Science: used in computer graphics, image processing, and machine learning to understand the properties of digital images and smooth the movements of on-screen objects.
6. Biology: used in many areas of biology, including population dynamics, physiology and physiology, genetics, epidemiology, and ecology to understand growth patterns and other phenomena.

This list is not exhaustive, it have many more application in many other fields as well.

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