# Calculus

**Calculus **is a subset of mathematics concerned with the study of continuous transition. Calculus is also known as infinitesimal calculus or “infinite calculus.” The analysis of continuous change of functions is known as classical calculus. Derivatives and integrals are the two most important ideas of calculus. The integral is the measure of the region under the curve, while the derivative is the measure of the rate of change of a function. The integral accumulates the discrete values of a function over a number of values, while the derivative describes the function at a given point.

Calculus, a branch of mathematics founded by **Newton **and **Leibniz**, study the pace of transition. Calculus Math is commonly used in mathematical simulations to find the best solutions. It aids us in understanding the changes between values that are linked by a purpose. Calculus Math is mostly concerned with certain critical topics such as separation, convergence, limits, functions, and so on.

Calculus Mathematics is generally divided into two types: **Differential Calculus** and **Integral Calculus**. Both differential and integral calculus consider the effect of a small shift in the independent variable on the equation as it approaches zero. Both discrete and integral calculus serves as a basis for the higher branch of mathematics known as **Analysis.**

## Table of Contents

### Differential Calculus

Differential Calculus deals with the issues of determining the rate of change of a parameter with respect to other variables. Derivatives are used to find the maxima and minima values of a function in order to find the best solution. The analysis of the boundary of a quotient leads to differential calculus. It is concerned with variables such as x and y, functions f(x), and the resulting variations in x and y. Differentials are represented by the symbols dy and dx. Differentiation refers to the method of determining derivatives. A function’s derivative is defined by dy/dx or f’ (x). It denotes that the equation is the derivative of y with respect to x. Let us go through some main topics discussed in simple differential calculus in the following articles:

- Introduction to Limits
- Formal Definition of Limits
- Strategy in Finding Limits
- Determining Limits using Algebraic Manipulation
- Limits of Trigonometric Functions
- Properties of Limits
- Limits by Direct Substitution
- Estimating Limits from Graphs
- Estimating Limits from Tables
- Squeeze Theorem
- Introduction to Derivatives
- Average and Instantaneous Rate of Change
- Algebra of Derivative of Functions
- Product Rule – Derivatives
- Quotient Rule
- Derivatives of Polynomial Functions
- Derivatives of Trigonometric Functions
- Power Rule in Derivatives
- Application of Derivatives
- Applications of Power Rule
- Continuity and Discontinuity
- Differentiability of a Function
- Derivatives of Inverse Functions
- Derivatives of Implicit Functions
- Derivatives of Composite Functions
- Derivatives of Inverse Trigonometric Functions
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Proofs for the derivatives of eˣ and ln(x) – Advanced differentiation
- Derivative of functions in parametric forms
- Second-Order Derivatives in Continuity and Differentiability
- Rolle’s and Lagrange’s Mean Value Theorem
- Mean value theorem – Advanced Differentiation
- Continuity and Discontinuity in Calculus
- Algebra of Continuous Functions
- Critical Points
- Rate of change of quantities
- Increasing and Decreasing Functions
- Increasing and Decreasing Intervals
- Separable Differential Equations
- Higher Order Derivatives

### Integral Calculus

The analysis of integrals and their properties is known as integral calculus. It is primarily useful for the following two functions: To compute f from f’ (i.e. from its derivative). If a function f is differentiable in the range under consideration, then f’ is specified in that range. To determine the region under a curve. **Differentiation **is the inverse of **integration**. As separation can be defined as the division of a part into several small parts, integration can be defined as the selection of small parts to form a whole. It is commonly used to calculate area.

A **definite integral** has a specified boundary beyond which the equation must be computed. The lower and upper limits of a function’s independent variable are defined, and its integration is represented using definite integrals. An **infinite integral** lacks a fixed boundary, i.e. there is no upper and lower limit. As a result, the integration value is always followed by a constant value. Following are the articles that discuss the integral calculus deeply:

- Tangents and Normals
- Equation of Tangents and Normals
- Absolute Minima and Maxima
- Relative Minima and Maxima
- Concave Function
- Inflection Points
- Curve Sketching
- Approximations & Maxima and Minima – Application of Derivatives
- Integrals
- Integration by Substitution
- Integration by Partial Fractions
- Integration by Parts
- Integration using Trigonometric Identities
- Functions defined by Integrals
- Indefinite Integrals
- Definite integrals
- Computing Definite Integrals
- Fundamental Theorem of Calculus
- Finding Derivative with Fundamental Theorem of Calculus
- Evaluation of Definite Integrals
- Properties of Definite Integrals
- Definite Integrals of Piecewise Functions
- Improper Integrals
- Riemann Sum
- Riemann Sums in Summation Notation
- Definite Integral as the Limit of a Riemann Sum
- Trapezoidal Rule
- Areas under Simple Curves
- Area Between Two curves
- Area between Polar Curves
- Area as Definite Integral
- Basic Concepts of differential equations
- Order of differential equation
- Formation of a Differential Equation whose General Solution is given
- Homogeneous Differential Equations
- Separable Differential Equations
- Linear Differential Equations
- Exact Equations and Integrating Factors
- Particular Solutions to Differential Equations
- Integration by U-substitution
- Reverse Chain Rule
- Partial Fraction Expansion
- Trigonometric Substitution
- Implicit Differentiation
- Implicit differentiation – Advanced Examples
- Disguised Derivatives – Advanced differentiation
- Differentiation of Inverse Trigonometric Functions
- Logarithmic Differentiation
- Antiderivatives