Calculus: Definition, Differential, Integral, Formulae, & Derivations

Calculus deals with continuous change and is one of the most significant branches of mathematics. It is also known as infinitesimal calculus. A majority of variables are time functions, such as velocity, which equals the change in distance with respect to time. Calculus is built around two key concepts: derivatives and integrals. 

Rate of change of a function is measured by its derivative, whereas the area under the curve of the function is measured by its integral. Derivatives explain the function at a specific point while integrals accumulate discrete values of a function over a range of values.

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Definition of Calculus

Calculus is a field of mathematics that was developed by Newton and Leibniz and is concerned with the study of change rates. Calculus is a branch of mathematics that focuses on differentiation, integration, limits, functions, and other key topics. Calculus Math is used to achieve optimal solutions in mathematical models. It helps in comprehending changes in values that are associated by a function. 

Mathematics of calculus is divided into two categories:

• Differential Calculus
• Integral Calculus

The influence on the function of a small change in the independent variable as it goes to zero is dealt with in both differential and integral calculus. The basis for the higher discipline of mathematics known as "Analysis" involves both differential and integral calculus. In contemporary physics, as well as science and technology, calculus mathematics is essential.

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Precalculus

In mathematics, precalculus is a subject that incorporates trigonometry and algebra and is aimed to prepare students for calculus. The study of complex mathematical topics such as functions and quantitative reasoning is the subject of precalculus. Precalculus covers a number of key subjects, including

• Functions
• Inverse Functions
• Complex Numbers
• Rational Function

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Basic Calculus

Differentiation and integration are the subjects of basic calculus. Limits and functions are at the heart of both notions. The cornerstone of advanced calculus is several ideas such as continuity and exponents. The two forms of calculus, "Differential Calculus" and "Integral Calculus," are explained in basic calculus. Differential calculus aids in determining a quantity's rate of change, whereas integral calculus assists in determining the quantity when the rate of change is known.

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Differential Calculus Basics

Differential Calculus is concerned with determination of a function's rate of change in relation to other variables. Derivatives are used to identify maximum and minimum values of a function in order to find the best solution. 

Study of a quotient's limit gives birth to differential calculus. It deals with variables like x and y, as well as functions like f(x) and the changes in the variables x and y. Differentials are represented by the symbols dy and dx. Differentiation is the process of determining the derivatives. dy/dx or f'(x) is the symbol for a function's derivative (x). The function is defined as the derivative of y with respect to the variable x.

Limits

Limit is useful for determining the degree of proximity to any value or approaching phrase. The limit formula is commonly used to represent a limit as,

Limx→cf(x)=A

This expression is read as “the limit of f of x as x approaches c equals A”.

Derivatives

Derivatives reflect the rate of change of one quantity in relation to another at any given time. A function's derivative is expressed as:

Limx→h[f(x + h) − f(x)]/h=A

Also check: Class 12 Mathematics Chapter 6 Applications Of Derivatives Important Topics

Continuity

A function f(x) is said to be continuous at a particular point x = a, if the following three conditions are satisfied:

f(a) is defined

limx→ax→af(x) exists

limx→a-f(x)=limx→a+f(x)=f(a)

Continuity and Differentiability

If a function is differentiable at any point, it is always continuous; however, the converse is not necessarily true.

Quotient Rule

The Quotient rule is a technique for calculating the derivative (differentiation) of a fractional function.

Chain Rule

The chain rule is the method for determining the derivative of a function's composition.

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Integral Calculus Basics

Integral calculus is the study of integrals and their properties. It is mostly useful for the following two purposes:

• To calculate f from f’ (i.e. from its derivative). If a function f is differentiable in the interval of consideration, then f’ is defined in that interval.
• To calculate the area under a curve.

Integration

Differentiation is the inverse of integration. Integration may be thought of as a collection of little components that come together to make a whole, similar to how differentiation can be thought of as separating a part into numerous small bits. It is commonly used to calculate area.

Also, check: Class 12 Mathematics Chapter 8 Applications Of The Integrals Important Topics

Definite Integral

A definite integral has a defined range of values within which the function must be computed. The lower and upper limits of a function's independent variable are given, and the function's integration is represented using definite integrals. The following is the definition of a definite integral:

 abf(x).dx = F(x)

Indefinite Integral

An indefinite integral without a specified boundary, i.e. there is no upper or lower limit. As a result, the value of integration is always accompanied by a constant value (C). It's written as:

f(x).dx = F(x) + C

Also read:

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Calculus Formula

The following six major categories of calculus formulae can be used to categorise them. Limits, differentiation, integration, definite integrals, differentiation application, and differential equations are the six broad formulae. These formulae are mutually beneficial.

Limits Formulas

Limits are used to approximate values to a certain number, and they can be set to zero or infinity.

Also check: 

Differentiation Formula

Differentiation Formulas are applicable to basic algebraic expressions, trigonometric ratios, inverse trigonometry, and exponential terms.

Integration Formula

 Integral formulas can be derived from differentiation formulas and are complementary to differentiation formulas.

Definite Integrals Formulas

 Infinite integrals have limits in addition to being the basic integral formulas. There is an upper and lower limit, and definite integrals, that are helpful in finding the area within these limits.

Application of Differentiation Formulas

Differentiation formulas are useful for approximation, estimation of values, equations of tangent and normals, maxima and minima, and for finding the changes of numerous physical events.

dy/dx = (dy/dt)/(dx/dt)

Equation of a Tangent: y - y1 = dy/dx.(x - x1)

Equation of a Normal: y - y1 = -1/(dy/dx).(x - x1)

Differential Equations Formula

Differential equations are higher-order derivatives and can be comparable to general equations. In the general equation, we have the unknown variable 'x' and here we have the differentiation dy/dx as the variable of the equation.

Homogeneous Differential Equation: f(λx, λy) = λnf(x, y)

Linear Differential Equation: dy/dx + Py = Q

General solution of Linear Differential Equation is y.e- ∫P.dx = ∫(Q.e∫P.dx).dx + C

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Advanced Calculus

Infinite series, power series, and other advanced calculus subjects are all essentially applications of the ideas of basic calculus topics like differentiation, derivatives, rate of change, and so on. Vector spaces, matrices, and linear transformation are some of the most significant domains for advanced calculus. Advanced Calculus enables us to obtain an understanding of a few key concepts, including:

• Quadratic forms
• Generalized Stokes theorem
• Vector fields as derivatives
• Integration of forms
• Multilinear algebra
• Continuous differentiability
• Tangent space and normal space via gradients
• Dual space and dual basis
• Critical point analysis for multivariate functions, etc.

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Applications of Calculus

Calculus is a mathematical branch that helps with:

• Analyzing a system in order to identify the best answer for predicting the future of any given function condition.
• Calculus concepts are important in everyday life, whether it's for solving the area of complex shapes, assessing survey data, automobile safety, company planning, credit card payment records, or determining the changing conditions of a system that influence us, for example.
• Economists, biologists, architects, medical specialists, and statisticians all use calculus. Architects and engineers, for example, employ distinct calculus ideas to determine the size and geometry of building structures.
• Calculus is used to simulate birth and death rates, radioactive decay, reaction rates, heat and light, motion, and electricity, among other things.