Differential calculus

From Simple English Wikipedia, the free encyclopedia

Differential calculus is a branch of mathematics dealing with derivatives of a function. The derivative of a function is a property (property = something that decribes and is part of) of a function which describes how fast a function changes when its argument changes. (what the function "takes" and uses to calculate a result). Differential calculus is extremely useful in Physics, as many processes and happenings in nature can be shown as a function of other things. It includes ways to calculate functions based on other functions using derivation (finding the derivative of a function) and theories which allow us to make statements and know facts about mathematical objects from derivatives.

1 Origin
2 Use in Physics
3 Mathematical meaning
- 3.1 Calculating derivatives
- 3.2 Derivatives of common functions

[edit] Origin

Calculus in general was invented by Sir Isaac Newton in the 17th century (the years 1600 - 1700) and developed by many including Gottfried Leibnitz, Augustin Louis Cauchy and many other mathematicians along the years. It was intended (made for) describing Physical occurences (things that take place in nature).

[edit] Use in Physics

Many occurences in Physics can be described as a function of something else, and so Differential calculus is used to describe and calculate those functions. Almost anything in Physics which can be measured or shown in numbers is a function, and we are often interested in its rate of change, which is described by its derivative. For example, we can use a function "place(t)" to mean the place of an object or its distance measured from a certain point as a function of the time, or simply where the object is at a given time. The rate of change of the place of an object is the object's speed, so we can call the function of its speed at any time "speed(t)". The rate of change of speed is called acceleration. We can call the function "acceleration(t)". Mathematically, "speed(t)" is the derivative of "place(t)" and "acceleration(t)" is the derivative of "speed(t)". Using calculus, we can tell an object's speed and accelleration just by knowing its place at any time.

[edit] Mathematical meaning

The mathematical definition of the derivative is the change in the value of function when its argument is changed by a bit compared to the change in the argument.

   Example: The derivative of the square function  $x 2$  is . It means that for example if
   at the point  $x = 5$  the value of the function is  $52 = 25$ , at a point which is changed by a bit,
   for example  $x = 5 + 0.1 = 5.1$ , the value of the function will be changed 10 times as much because the derivative
   is equal . So, the value of  $x 2$  at point 5.1 would be close to
   . Using the derivative, we can know that  $5.1 2$  is something very
   close to 26. If we calculate exactly,  $5.1 2 = 26.01$ . The result is very close indeed.

Differential calculus also includes many useful theorems (statements which are always true in Math) about functions.

  Example: A very simple theorem stated that when the derivative of a function is bigger than 0, adding to the argument
  will add to the value of the function and not subtract from it. If we look at the previous example, the derivative 
  equals 10, which is bigger than 0. So, when we added 0.1 to the argument (5.1 instead of 5) the value of the function 
  became larger: 26 instead of 25.