Calculus

Calculus is used to study and understand changes. The changes can be variations in action, speed, or time. The important functions of calculus are differentiation, integration, and limits. It was developed by Newton and Leibniz.

Calculus is a branch of mathematics, which deals with the study of motion. It includes derivatives, integrals, limits, and so on. The need to calculate the area of complex shapes like spirals led to the development of calculus. 
 

In earlier times, people used calculus to find the area and volume of an object. For that, they used exhaustion and inverse heuristic, a precursor to limits and similar to integrals. Archimedes, Ibn al-Haytham, Bhaskara II, and Cavalieri were some ancient mathematicians who used techniques related to calculus.

The attribution for the development of calculus is given to Isaac Newton and Gottfried Wilhelm Leibniz. They used it to find the curvature and solve complex problems. Newton's work was based on derivatives for calculating the rate of change. During the same time, to calculate the area under curves, Leibniz used integral calculus.

Today, we use it in the fields of economics, music, weather forecasting, and many more.  

Based on the change and mathematical analysis, calculus can be divided into four parts. 

• Differential Calculus 
• Integral Calculus
• Multivariable Calculus
• Vector Calculus
   

Differential calculus is used to find the rate of change of a quantity with respect to another. The key concepts are limits, continuity, and slope of a curve.

To find the quality when the rate is known, integral calculus is used. Integral is the inverse process of differentiation. We use it to calculate the area and volume. 

It is used when the function of two or more variables is calculated. It is an extension of differential and integral calculus. 

Vector calculus is about the use of differentiation and integration of vector fields in Euclidean space. 

The integral is the inverse process of the differential. The fundamental theorem of calculus is all about the connection between them. There are two theorems that are the first and the second fundamental theorem of calculus. 
 

In calculus for each function, we have different formulas. And students should remember and use the correct formula. For a better understanding of calculus, it's important to remember all the basic formulas. 

d/dx(xⁿ)=nx^n-1

d/dx(C)=0  → C is the constant

d/dx(Cf)=C.d/dx(f)

d/dx(f±g)=d/dx(f) ± d/dx(g)

d/dx(fg)=d/dx(fg) +d/dx(gf)

d/dx(fg)=g(d/dx)(f) - fd/dx(g)/g²

d/dx(sin x)=cos x

d/dx(cos x)= - sin x

d/dx(tan x)= sec²x

d/dx(cot x) = -cosec²x

d/dx(secx)=sec x tan x

d/dx(cosec x) = -cosec x cot x

d/dx(a^x) = a^x In a

d/dx(e^x) =e^x

d/dx(In x) = 1/x

d/dx(log_ax) = 1/x In a

1dx = x +C

a dx =ax + C

1xdx = In |x| + C

11-x2dx = sin-1x + C

11+x2dx = tan-1x + C

1|x| 1-x2dx = sec-1x + C

sin x dx = -cos x + C

cos x dx = sin x + C

sec x (tan x) dx = sec x + C

csc x (cot x) dx = -csc x + C

sec2 x dx = tan x + C 

csc2 x dx = -cot x + C 

e2 dx =ex+ C

a2 dx = axIn a + C 

1xdx = In |x| + C

loga x dx = x In x -xIn a + C

We have talked a lot about calculus. We know that we use it in our daily life. Now let’s see the applications of calculus.

Physics: In Physics, calculus is used to calculate motion, force, change, energy and many more. The speed of a vehicle, rocket, or even the light can be calculated. 

Engineering: From building bridges to developing circuits, we use calculus in different fields of engineering.   

Economics: Calculus is used to study the market and its behavior. To understand the minimum cost and maximum profit 

Medicine: To study the growth of bacteria and infectious diseases calculus is used. This data is used to treat cancer and diagnose patients. 
 

We have already learned what is calculus in math. Calculus is a broad topic that includes limits, derivatives, integrals, and many more. Let’s learn some tips and tricks to make it easier for students to learn
 

Understand fundamental concepts like limits and continuity 

Understanding the fundamental concepts is the best way to learn calculus. Derivatives, integrals, their relationship, limits, and continuity are the few fundamental concepts. 

Practice derivatives and integral rules repeatedly

Students can learn calculus by remembering the basic rules. Regular practice will help students remember the rules. 

Visualize problems using graphs

Students can use visual aids like graphs to understand the problem. To calculate the change, slope, area, cures, and so on, students can use graphs. 

Verify using reverse operations
When working on derivatives and integrals, students can verify the answer using reverse operations. As integration is reverse operation of differential.