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Differentiation and Integration Formulas





Mathematics is a very interesting subject. There are many formulae that are used in mathematics to solve problems. The differentiation and integration formula is one among them.
Both these are used in the field of calculus. They are very important part of calculus and form the backbone of calculus. 
To understand the concept of differentiation one must also understand the concept of function. There are inputs to the function. 
These inputs might change over a period of time. 

So, differentiation in Calculus can be used to find the derivative of a function. 
This derivative can be used as a measure of knowing how much the function changes with the changes in the inputs that are given to the function. So, one can use the concept of differentiation to find the derivative.  Various functions are inputs to the differentiation formula. 
There is a list of formulae that could be used for the purpose of finding the differentiation of a function. 
The differentiation of an exponential raised to the power of ‘x’ is the same exponential raised to the power of ‘x’. This is one of the formulae. 

There are several formulae and these depend on the function that is given as the input.
Integration on the other hand can be thought of the inverse process of differentiation. 
There can be both definite and indefinite integrals. For the definite integrals limits are defined but for the indefinite integrals no limits are defined.

Basically when the limits are used the integration is used to represent the area that is bounded by the function, over which integration is performed. This area is present in the x-y plane. The differentiation and integration formula must be studied in order to solve the problems based on differentiation and integration. Both these tools are very important tools of Calculus and have various applications in different fields.  

When the final limits of the integration are to be applied then the value of the upper limit is put into the function first and then the value of the function is found out. Similarly the value of the lower limit is put into the function and the value of the function is found out. 
Now the difference between the value obtained by putting the upper limit value in the function and the value obtained by putting the lower limit value in the function is found out. This is the value of the finite integral of the function over which integration is performed.

In integration there are concepts of line integral and surface integral. The line integral can be performed over functions that can have two or three variables. 
A variable is one whose value changes over and is not fixed or constant. 
The surface integral can be performed over a piece of space that is present in the three dimensional space. 
This is the basic difference between the two types. This has to be kept in mind while performing the operation. 

The differentiation and integration formula must be learnt thoroughly and only then they can be effectively used to solve the problems. 
Sometimes double or triple integrals can also be performed. They are a bit more complex than the simple integrals but the procedure remains the same. The differentiation of a constant number is zero. 
This is one of the formulae of differentiation.
  
An example of differentiation can be given from the field of science. If an object is moving, the rate of change of its position is found with respect to time then the value obtained is nothing but the velocity of the moving object. 
So, this is one of the simple applications.