Introduction to Differentiation and Integration

We know that Mathematics is one of the important subjects that each student has to learn. Mathematics is not only helpful in academics; it is useful in many real-life scenarios also. It is a subject which is incredibly important for most of their lives. Mathematics is not about just numbers; it is all about patterns. Many fields like designing, engineering, marketing, finance require good Math skills.

In Mathematics, we will come across many concepts. All the ideas are equally important from the basic concept of counting to the complex concepts like differentiation and integration in calculus. Here, we are going to discuss the introduction of differentiation and integration along with its applications.

Generally, calculus can be broadly classified into two types, namely:

• Differential Calculus - It deals with finding the changes in the given function. It breaks the functions into small pieces to determine its changes.

• Integral calculus - It integrates the smaller functions to find the whole.

Differentiation

Differentiation is the process of determining the derivative of a function. The fundamental concept of differentiation is finding the derivative; In other words, differentiation is a method of finding the function’s slope or change rate. It means that the rate at which the output value will change for the input value. The derivatives generally define the slope of the given function at any point. It is the measurement for the steepness of the graph function. The derivative of the function is generally represented using the notation “dy/dx”. 

For example, if f(x) is the given function, the derivative is represented by f’(x) = dy/dx. It represents the derivative of the variable y with respect to the variable x. Here “x” is the independent variable, and “y” is the dependent variable.  

Applications:

• It helps to calculate the temperature change.

• It helps to determine profit and loss.

• It is used to derive many physics equations.

• It is helpful to calculate speed and distance.

Integration

Integration is the reverse process of differentiation. It is used to find the volume, area and central points. The area under the curve of the given function can be easily computed using integration. The key concept involved in the integration is the “Integral”. Finding integral is the reverse process of finding the derivative. It is the collection of many small parts to make it whole. The symbol “∫” represents the integral. The integral can be classified into:

Definite Integral: If the integral has upper and lower bounds, then it is known as definite integral

Indefinite Integral: If the integral is not specified with upper and lower bounds, it is indefinite integral.

There are different methods of integrations, which helps to find the integral of a function. Some of the methods are integration by substitution, integration by parts, integration by partial fraction, etc.

Applications:

• Integration is used to find the area under the curve, and area between two curves.

• It is used to find kinetic energy, the centre of mass, distance, velocity, etc.

• It helps to find the average value of a function.