Calculus is a branch of mathematics that deals with the study of rates of change and the accumulation of small changes to determine the behavior of functions. It is divided into two main branches: differential calculus and integral calculus.

Differential Calculus:

Differential calculus deals with the study of rates of change of functions. It involves finding the derivative of a function, which is the rate of change of the function at a particular point. The derivative of a function f(x) is denoted by f’(x) or dy/dx.

The formula for finding the derivative of a function is:

f’(x) = lim(h->0) [f(x+h) - f(x)]/h

This formula is known as the limit definition of the derivative. It involves taking the limit of the difference quotient as h approaches zero.

There are several rules for finding derivatives of functions, including:

1. Power rule: If f(x) = x^n, then f’(x) = nx^(n-1)

2. Product rule: If f(x) = u(x)v(x), then f’(x) = u’(x)v(x) + u(x)v’(x)

3. Quotient rule: If f(x) = u(x)/v(x), then f’(x) = [u’(x)v(x) - u(x)v’(x)]/v(x)^2

4. Chain rule: If f(x) = g(h(x)), then f’(x) = g’(h(x))h’(x)

Integral Calculus:

Integral calculus deals with the study of accumulation of small changes to determine the behavior of functions. It involves finding the antiderivative of a function, which is the inverse operation of differentiation. The antiderivative of a function f(x) is denoted by ∫f(x)dx.

The formula for finding the antiderivative of a function is:

∫f(x)dx = F(x) + C

where F(x) is any function whose derivative is f(x), and C is the constant of integration.

There are several rules for finding antiderivatives of functions, including:

1. Power rule: If f(x) = x^n, then ∫f(x)dx = (1/(n+1))x^(n+1) + C

2. Integration by substitution: If f(x) = g’(x)h(g(x)), then ∫f(x)dx = ∫h(u)du, where u = g(x)

3. Integration by parts: If f(x) = u(x)v’(x), then ∫f(x)dx = u(x)v(x) - ∫v(x)du

4. Trigonometric integrals: If f(x) contains trigonometric functions, then there are several formulas for finding antiderivatives, such as:

∫sin(x)dx = -cos(x) + C

∫cos(x)dx = sin(x) + C

∫tan(x)dx = ln

sec(x)

+ C

Calculus is a powerful tool for solving problems in physics, engineering, economics, and many other fields. It provides a way to model and analyze complex systems by understanding how they change over time.