Introduction to Derivatives

Limits and Derivatives are the foundation stones of calculus. Just as limits help in measuring exponential functions, derivatives help study the nature of their change. Derivatives have addition, subtraction, multiplication, and division rules, which allow us to bypass the variable’s value and yet find desired measurements. These applications are indirectly used in various fields such as Engineering, Astronomy, Physics, Information Technology, Biotechnology, Agrosciences, etc. A function is continuous in its motion. The meaning of derivative can be defined as the momentary change in the constant motion of an exponent. The derivative of a function is a small unit of it. It is denoted as dx. The unit dx can be an integer but never zero. 

INTRODUCTION

A derivative is a fundamental part of calculus. It can be explained as the instant by the instant varying rate of change of the function of a variable to an independent variable.

When scientists want to study a dynamic system, i.e., a system whose components are constantly changing, they use calculus. For example, if they wish to study a particular molecule of water in the ocean, they will obtain its rate of change compared to another water molecule. This is the meaning of derivatives in calculus. This information can be further used by the scientists to apply in a differential equation and then use integration to study the behavior of the above system in different conditions.

Thus it is a very useful tool for making calculations and predictions. Derivatives also make it possible for scientists to test their theories mathematically. Once mathematically proven, a theory is ready to be tested practically. This is why derivatives are considered so important in calculus.

Types of Derivatives

Derivatives can be put into two categories;

• First-Order Derivatives

A first-order derivative tells us about the direction of change of a function. It can be increasing, which is called exponential growth, or decreasing, which is called exponential decay. It is measured with time as a constant. Hence, it can also be called the instantaneous rate of change. In a graph, an incline or decline shows the first-order derivative. A real-life application of a first-order derivative is to maximize the power delivered to circuits. It maximizes the power delivered to a load in electronic circuits.

• Second-Order Derivatives

The second-order derivative tells us more about the curve of the slope representing the movement of the variable. It depends upon the concavity of the curve. The concavity shows the rapidness of the change of a function. There can be two cases under the second-order derivatives, concave-up or concave-down.

Limits and Derivatives

Limits and Derivatives are the foundation stones of calculus. Just as limits help in measuring exponential functions, derivatives help us study the nature of their change. The concepts of limits are combined with derivatives to understand functions better. 

Derivatives of a Function

A function is continuous in its motion. A derivative is a momentary change in the continuous motion of an exponent. Sir Isaac Newton gave birth to concepts like derivatives while propounding the theories of motion. The derivative of a function is a small unit of it. It is denoted as dx. The unit dx can be an integer but never zero. Using this small unit, derivative formulae are created.

Properties of Derivatives

Derivatives have rules of addition, subtraction, multiplication, and division which allow us to bypass the value of the variable and yet find desired measurements. For example, to find the derivative of the sum of two functions, the value of the function is not needed. The sum of the derivatives of the two variables will be equal to the derivative of the sum. 

Derivatives vs. Integrals

The integral of a function is also related to the slope of the exponential function’s curve, but they are different from derivatives. Integrals explain the area of a slope of the function, while derivatives focus on the shape of the curve. 

Applications of Derivatives

Derivatives have many applications in conceptual mathematics such as to determine;

• the Rate of Change of a Quantity 
• the Approximation Value 
• the equation of Tangent and Normal to a Curve
• the Minimum and Maximum Values of algebraic expressions

Important mathematical formulae such as Integration formulae, Differential Equations, etc., are based on derivatives. Trigonometry also plays a significant role in these formulae as the slope’s angles have to be calculated.

Applications in other fields

These applications are indirectly used in various fields such as Engineering, Astronomy, Physics, Information Technology, Biotechnology, Agrosciences, etc. In everyday life, the derivative can tell you at which speed you are driving or help you to understand the fluctuations in a stock market. In machine learning, derivatives are used for task optimization. Architects use it too. They could have used geometry if buildings were made of ideal squares and triangles. However, structures involve curves and strange angles. An essential part of architecture is figuring out how to minimize or maximize an aspect of the building. For example, to maximize window size in a building, an architect can find an optimum space by using a derivative (in other words, where the slope is zero) to create the most window space possible based on the proportions and aesthetics of the building. The newer applications of derivatives include Derivative Trading, which has become increasingly popular in the twenty-first century.

CONCLUSION

Derivatives are the first step to Calculus. Before learning differentiation and integration, fundamental concepts like derivatives and limits must be understood. The role of derivatives in mathematics will help you understand their importance in fields like engineering. Even in other new-age fields, derivatives have become indispensable. Basic knowledge of derivatives will make you comfortable with more complex concepts of differential calculus and later integration.