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Calculus Formulas

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Last updated date: 05th Mar 2024
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Introduction to Calculus

Calculus is known to be the branch of mathematics, that deals in the study rate of change and its application in solving equations. During the early Latin times, the idea of Calculus was derived from its original meaning “small stones” as means of computing a calculation of travelling distance or measuring and analyzing the movement of certain objects like stars from one place to another with their space covered in real-time. Apart from this, Calculus was used as a means of computation in various other fields also. 

 

For Eg: To gain a better understanding of the movement of a car about its speed at different intervals and time taken for travelling, Calculus can be used for easy computation. In other words, Calculus is a significant tool for measuring the tendency of fluctuations considering the nature of an object. 

 

Principles of Calculus

  1. Limits and Infinitesimals: Infinitesimals refer to extremely small digit numbers such as values between 0 and 1. In other words, numbers that are less in value compared to a positive real number are called Infinitesimals. The concept of Calculus formulas was developed at first to compute such small values and thus, it can manipulate certain limits and principles for infinitesimals. 

  2. Differential Calculus: Differential Calculus is one of the branches of calculus which is discussed further in detail. The process of deriving a derivation out of a function is termed Differentiation. Derivatives help in producing different functions and their output usually by squaring the values of the given digits. 

  3. Leibniz Notation: The term Leibniz Notation is coined after the great Mathematician and German Philosopher Gottfried Wilhelm Leibniz. Leibniz Notation utilises the symbols of “dx” and “dy” to find the exact value of derivatives. The values of “dx” and “dy” are taken as assumptions to compute the further calculation of any given function. 

  4. Integral Calculus: Integral Calculus is another branch of Calculus along with Differential Calculus. This particular concept is discussed more in detail further. In simple words, it is a study of the internal properties of a given function and its application in different fields. Integration refers to the idea of deriving the value of an integral. There are two types of integrals studied on a general basis namely; Indefinite Integral and Definite Integral. Indefinite integrals are denoted by “f” as they represent a number of functions that are non-constant in nature. On the other hand, definite integral functions consist of a series of functions that are constant in nature. 

  5. Fundamental Theorem: The fundamental theorem of Calculus refers to the relationship between and the derivatives and their integrals. 

 

Differential calculus and integral calculus are the two major branches of calculus. Differential calculus formulas deal with the rates of change and slopes of curves. It can be used to study and examine the trend of a given graph and highlight its maximum and minimum values of curve points. The rate of change can be studied considering its quantity using Differential Calculus.  

 

For Eg: In our day-to-day life, the method of Differential Calculus can be used to scrutinize the change in rising and fall of temperature of a given place and at a given point of time. This helps to study the Weather Conditions of different places and compare them with one another later on. In the same way, the statistics related to the growth of a business entity can be compared by analyzing their profit and loss using Differential Calculus. As well, it can be used to measure the rate of change in speed of a vehicle travelling different miles or kilometers at given hours. 

 

Integral Calculus deals mainly with the accumulation of quantities and the areas under and between curves. In calculus, we use the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. These fundamentals are used by both differential and integral calculus. Unlike Differential Calculus which studies the rate of change and the number of different objects, Integral Calculus examines the nature of a particular object in the context of its speed, time-taken, and movement. It analyzes the internal properties of a change and computes advanced calculations of the area covered its length and volume.  

 

In other words, we can say that in differential calculus, an area splits up into small parts to calculate the rate of change. Whereas in integral calculus small parts are joined to calculate the area or volume and it is the method of reasoning or calculation. On one hand, Differential Calculus is used to compute the calculation of the rate of change, for instance, the velocity of the speed of a given object at a given point of time, then, on the other hand, Integral Calculus is used to study the area covered by that respective object and the time taken during the process. While Differential Calculus studies the minimum and maximum value of a given curve and highlights the curve point, Integral Calculus studies the area that has been covered under the respective curve. 

 

For calculating very small quantities, we use Calculus. Initially, the first method of doing the calculation with very small quantities was by infinitesimals. For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, etc.

 

In other words, its value is less than any positive real number. According to this point of view, we can say that calculus is a collection of techniques for manipulating infinitesimals. In calculus formulas, the symbols dx and dy were taken to be infinitesimal, and the derivative formula dy/dx was simply their ratio.

 

What are the Limits?

Suppose we have a function f(x). The value, a function attains, as the variable x approaches a particular value let’s say suppose a that is., x → a is called its limit. Here, ‘a’ is some pre-assigned value. It is denoted as

lim x→a f(x) = l

  • The expected value of the function f(x) shown by the points to the left of a point ‘a’ is the left-hand limit of the function at that point. It is denoted as limx→a−  f(x).

  • The point to the right of the point that is ‘a’ which generally shows the value of the function is the right-hand limit of the function at that point. It can be denoted as limx→a + f(x).


Limits of functions at a point are the common and coincidence value of the left and right-hand limits.


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The value of a limit of a function f(x) at a point a that is, f(a) may vary from the value of f(x) at the point ‘a’.

 

Math Limit Formula List

1

\[\lim_{x \to 0}sin x = 0\]

2

\[\lim_{x \to 0}cos x = 1\]

3

\[\lim_{x \to 0}sin \;x/x = 1\]

4

\[\lim_{x \to 0}log\;(1+x)/x = 1\]

5

\[\lim_{x \to 0}log\;e^{x} = 1\]

6

\[\lim_{x \to e}log\;e^{x} = 1\]

7

\[\lim_{x \to 0}log\;e^{x} - 1/x = 1\]

8

\[\lim_{x \to 0}log\;a^{x} - 1/x = log_{e^{a}}\]

 

Differential Calculus & Differential Calculus Formulas

The study of the definition, properties, and applications of the derivative of a function is known as Differential calculus. Differentiation is the process of finding the derivative. Here are some calculus differentiation formulas by which we can find a derivative of a function.

 

Differential Calculus Formulas

1

\[\frac{\mathrm{d} }{\mathrm{d} x}\] (c) = 0

2

\[\frac{\mathrm{d} }{\mathrm{d} x}\] (x) = 1

3

Constant Multiple Rule : \[\frac{\mathrm{d} }{\mathrm{d} x}\] c(x) = 1

4

Sum or Difference Rule : \[\frac{\mathrm{d} }{\mathrm{d} x}\][f(x)+g(x)] = f’(x)+g’(x)

                                            \[\frac{\mathrm{d} }{\mathrm{d} x}\] [f(x)-g(x)] = f’(x)-g’(x) 

5

Power Rule : \[\frac{\mathrm{d} }{\mathrm{d} x}\](xn) =nxn-1 , where n is any real number.

6

Natural Exponential Rule: \[\frac{\mathrm{d} }{\mathrm{d} x}\] (ex) = ex

7

Product Rule : \[\frac{\mathrm{d} }{\mathrm{d} x}\] [f(x).g(x)] = f’(x)g(x)+f(x)g’(x)

8

Quotient Rule : \[\frac{\mathrm{d} }{\mathrm{d} x}\] [f(x)/g(x)] = f’(x)g(x)-f(x)g’(x) / [g(x)]2

9

Chain Rule : ( f∘g )( x ) equals f ′ ( g( x ) )·g′( x )

 

Integration Formulas

The branch of calculus where we study about integrals, accumulation of quantities, and the areas under and between curves and their properties is known as Integral Calculus. Let’s discuss some integration formulas by which we can find integral of a function. Here’s the Integration Formulas List

 

∫ xn dx

\[\frac{x^{n+1}}{n+1}\]  + C, where n ≠ -1

∫ sin x dx

- cos x + C

∫ cos x dx

sin x + C

∫ sec2 x dx

tan x + C

∫ cosec2 x dx

-cot x + C

∫ sec x tan x dx

sec x + C

∫ cosec x cot x dx

-cosec x +C

∫ ex dx   

e+ C

∫ 1/x  dx   

ln x+ C

∫ \[\frac{1}{1+x^{2}}\] dx

arctan x +C

∫ ax dx  

\[\frac{a^{x}}{ln\; a}\] + C

 

Questions to Be Solved

Question 1. Solve the Following Definite Integral

\[\int_{2}^{4}x^{2}\] dx

Solution

\[\int_{2}^{4}x^{2}\] dx

\[\int_{2}^{4}x^{2}\] dx = \[[\frac{x^{3}}{3}]_{2}^{4}\]

= \[\frac{64}{3}\] - \[\frac{8}{3}\]

= \[\frac {56}{3}\]

= 18.66 

FAQs on Calculus Formulas

1. What are the Calculus Formulas?

Calculus formulas basically describe the rate of change of a function for the given input value using the derivative of a function/differentiation formula. It is a process of studying a continuous change and computing the respective calculations of a given object and its nature for the same. The process of finding the derivative of any given function is known as differentiation. Whereas, the process of studying the internal properties of a given object is called Integration. 

2. What is Basic Calculus?

The basic function of Calculus is to study change and calculate the same. In basic calculus, we learn rules and differentiation formulas, which is the method by which we calculate the derivative of a function, and integration, which is the process by which we calculate the antiderivative of a function. There are two main branches of Calculus: (1) Differential Calculus and (2) Integral Calculus. Major calculations are carried out with the given methods. 

3. What are the 4 concepts of Calculus?

Outline of calculus. Calculus is known to be a branch of mathematics that is focused on limits, functions, derivatives, integrals, and mainly infinite series. This subject does constitute a major part of contemporary mathematics education. On one hand, the derivatives focus on the amount of change of a given object, the Integration on the other hand focuses on its internal properties such as the area covered or length of time taken during the movement of change. 

4. What is the study of Calculus?

Calculus is the study of how things change. It provides a framework for modeling systems in which there are changes and a way to deduce the predictions of such models. Calculus was used as a means of Computation in various other fields also. For more information, you can register on Vedantu.

5. Who is the Father of Calculus?

Calculus, is the branch of Mathematics that is known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Gottfried Wilhelm Leibniz and Isaac Newton were the ones who independently developed the theory of infinitesimal calculus in the later 17th century. It stands as one of the most reliable means of Computation of calculation to this date and it is well known as the branch of Mathematics.