Mathematics is an abstraction and generalization of basic concepts, one of the most basic of which is counting. Arithmetic is the study of numbers, each of which represents unknown objects. Algebra is the next step, the study of variables, a representation of unknown numbers. Calculus continues to build on the foundations of algebra. It is the study of equations, each of which represent relationships between unknown variables.

There are two basic branches, differential and integral calculus, on which the higher branches of calculus are based:

Differential calculus starts with the slope of a curve. An attempt is made to find a function, which gives the slope of the given curve at any one point. This function is called the derivative of the given function. Some functions are easy to find a derivative for and others are quite hard. Some are practically impossible.

Examples:

• If $f(x)=x$ then $f'(x)=dx$
• If $y=x^3$ then $dy=3x^2 dx$
• If $y=log(x)$ then $dy=\frac{1}{x} dx$

Integral calculus studies the area beneath a function. It can be counter intuitive to imagine areas beneath a curve, which is changing slope constantly, but once we discover functions that will give us that area at any point we come to an amazing relationship between differential and integral calculus. They are inverse functions. Taking a derivative undoes the integral and vice versa. This relationship helps us to expand our base of integrals very quickly since integration is not as easy or basic as differentiation at first glance.

Examples:
• $\int x=\frac{x^2}{2} +C$
• $\int x^3=\frac{x^4}{4}+C$
• $\int \frac{1}{x}=log(x) +C$
• $\int_0^3 x^2=9$

Much of the rest of basic calculus is expanding our vocabulary of functions and learning methods with which to find more derivatives and integrals.