Calculus is a branch of mathematics that deals with the study of rates of change and accumulation of quantities. It has two main branches: differential calculus and integral calculus.

1. **Differential Calculus**: This branch focuses on the concept of derivatives, which represent the rate of change of a function. It’s used to analyze the behavior of functions and solve problems involving rates of change, such as velocity, acceleration, and optimization.

Example: Consider a function \( f(x) = x^2 \). The derivative of this function, denoted \( f'(x) \) or \( \frac{df}{dx} \), gives us the rate of change of \( f(x) \) with respect to \( x \). In this case, \( f'(x) = 2x \), indicating that the rate at which \( f(x) \) changes with respect to \( x \) depends on the value of \( x \). For example, when \( x = 3 \), \( f'(3) = 2 \times 3 = 6 \), meaning the function is changing at a rate of 6 units for every unit increase in \( x \).

2. **Integral Calculus**: This branch deals with the concept of integrals, which represent the accumulation of quantities and the area under curves. It’s used to find the total change or accumulated quantity over an interval.

Example: Suppose we want to find the area under the curve of the function \( f(x) = x^2 \) from \( x = 0 \) to \( x = 2 \). This can be represented as the definite integral \( \int_{0}^{2} x^2 \, dx \). Evaluating this integral gives us the area under the curve, which in this case is \( \frac{2^3}{3} – \frac{0^3}{3} = \frac{8}{3} \).

Overall, calculus provides powerful tools for understanding and analyzing the behavior of functions, solving real-world problems involving rates of change and accumulation, and making predictions in various fields such as physics, engineering, economics, and biology.