The purpose of this article is to give a basic introduction to the world of calculus and its uses in mathematics, all of the sciences and the financial markets, to name a few. Put simply, calculus is the study of limits and infinite series. By putting these 2 things together, we have derivatives and integrals.
Derivatives give the rate of change in the value of a function, with respect to one of its variables. For example, we have the derivative with respect to time of position, which gives us velocity. The derivative with respect to time of velocity gives us acceleration.
Formally, and derivative of a function is the ratio of the change in the value of a function and an infinitesimally small change in a variable. Hence, we are taking the limit of the change in the value of the function as the change in the value of the variable goes to zero.
The exact method in which functions are differentiated (the method by which we find the derivative of a function) varies from function to function. For example, for polynomials in 1 variable (eg. x^3) we simply multiply by the power, and then reduce the power by one. So, the derivative of x^3 is 3x^2. We can also find the derivative of a sum or multiple of functions, and the derivatives are individually summed or multiplied as required. So for 3x^2 + 2x we have a derivative of 6x + 2. Another useful way of thinking about differentials is that they give the gradient of a plot of the function.
An integral is the limit of an infinite sum. Furthermore, intergrating the derivative of a function gives us the function itself (plus a constant to be determined by the "boundary conditions"). So to integrate a polynomial we increase the power by one and then divide by the new power.
A useful property of integrals, is that they can be thought of as giving the area under a plot of the function. For example, the integral of y = 2x^2 between x=0 and x=2 gives the area under the graph of the function between x = 0 and x = 2. Notice that with this integral I have declared the limits of the integral (namely x = 0 and x = 2); this type of integral is known as a definite integral.
An indefinite integral is one for which we have no integration limits, and the integral must hold for all x. We must first notice that the differential of any constant gives zero. This can be seen from the fact that x^0 = 1 so any constant term can be expressed as a multiple of x^0 which differentiates to zero. Graphically this can be seen by the face that any constant value is represented by a straight horizontal line, for which the gradient is zero. Hence, when we intergrate a function we must always add a "plus constant" term which can be calculated if we know some additional information about the function. For example, we could be given that at x = 0, y = 1. Hence for y = 2x, we integrate to y = x^2 + C. We know that y = 1 for x = 0, so C must be 1.
If you understand the basics of this article, you will be well prepared to further your knowledge into the calculus of other functions and working in more than one dimension / variable. A natural progression is to then study differential equations, and you'll find their uses particularly powerful in numerous fields.