Calculus Formulas

Calculus is known to be the branch of mathematics, which deals in the study rate of change and its application in solving the equations. 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.

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.

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 calculates the area or volume and it is the method of reasoning or calculation.

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.

(Image to be added soon)

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

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

= 56/3

= 18.66 

FAQ (Frequently Asked Questions)

1. What is the Calculus Formulas?

Calculus formulas basically describes the rate of change of a function for the given input value using the derivative of a function/differentiation formula. The process of finding the derivative of any given function is known as differentiation.

2. What is the Basic Calculus?

In basic calculus, we learn rules and differentiation formula, 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.

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.

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.

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 indefinitesimal calculus in the later 17th century.