Technology

# Applications and the History of Differentiation

In general, differentiation and integration are the chapters that students run away from. But once you understand the basic concept on which these topics are based, things become crystal clear to you. The process of obtaining the derivative of a given function is called differentiation. We use the concept of differentiation to find out the minutest rate of change of a given function based on one of its variables, the other variable being constant. Differentiation is very useful in real life. We will discuss the applications of differentiation as well as its history in detail. But before that, let us understand in brief differential equations.

## Origin of the Concept of Differentiation

The concept of differentiation dates back to 200-300 BC. Famous Greek mathematicians like Euclid, Apollonius of Perga, and Archimedes developed the concepts around it. Indians have made contributions in this field too. The famous mathematician, Aryabhatta used the concept of infinitesimals while studying the orbit of the moon. This was the first time when infinitesimals were used to study the rates of change. Bhaskara II of the Shaka era made significant developments in the area to compute the rate of changes. Many important notions of calculus can also be found in Rolle’s theorem.

The developments that took place in the modern era are mainly credited to famous scientists Gottfried Wilhelm Leibniz and Issac Newton. They provided various approaches to differentiation. The fundamental theorem of calculus was developed by them which brought them to the limelight. After the development of the fundamental theorem of calculus, all the other developments that took place before became obsolete. Many scientists of the seventeenth century contributed to the field of differentiation. Scientists like Augustin Louis Cauchy, Karl Weierstrass, and Bernhard Riemann made significant developments in this field during the nineteenth century.

## Various Applications of Differentiation

There are various applications of differentiation which are discussed below:

• To measure the rate of change of any given quantity: It is one of the most commonly used applications of differentiation. When we need to find the rate of change of the volume of a cuboid with respect to its decreasing sides, we use the concept of differentiation.
• To calculate the maxima and the minima: We use the concept of differentiation to locate the lowest or the highest point of any given curve.
• To calculate the profit and loss: Differentiation is used to calculate the profit and loss of companies. It is also used in the calculation of the equilibrium point or the break-even point.
• To calculate magnitude: It is used in calculating the magnitude of earthquakes or the magnitude of body temperature.
• The concept of differentiation is widely used in the study of complex topics of the subject of physics.

## What Do You Mean by Differential Equations?

Those equations that have the derivative of a function that is not known are known as differential equations. The concept of differential equations is very relevant in the study of various other subjects like biology, engineering, physics, and so on. We study differential equations mainly for the purpose of studying the solutions that satiate the given equation and the different properties of the solutions. Differential equations can be classified into two categories: partial differential equations and ordinary differential equations. We use ordinary differential equations in various fields. It is used in the calculation of the motion of an object, the flow of electricity and it is also used in the explanation of the concept of thermodynamics. The medical sector uses it to check the positive or the negative growth of any particular disease with the help of graphical representation.

If you want to learn more about the concepts of differentiation in detail and in a fun and interesting manner, you may visit the website of Cuemath and learn math, the Cuemath way.