What is Calculus? If this question is troubling you, then fear not! You are not alone. To truly understand Calculus, NCERT books won’t suffice. Our expert mathematics faculty from the best IIT-JEE Coaching in Kerala have curated this blog exploring what calculus is: the idea of integration and differentiation simplified for your reference.
Calculus is the branch of mathematics that studies change; in a way, Geometry is the study of shapes, and Algebra is the study of generalizations of arithmetic operations. It’s the mathematical art that helps you see patterns in change and motion. If you’re wondering how to do integration and differentiation, you are reasoning through the logic of variations and time. It helps to create quantitative models of change to foresee and analyze the consequences. Calculus was developed in the late 17th century by Isaac Newton and Gottfried Leibniz. Newton developed Calculus for his work on the Laws of Motion and Gravitation, which helped him prove that ellipses were actually sections of cones. The concepts and techniques of Calculus have great applications in science, engineering, and mathematics, even in 2025. Put your seatbelts on! Let’s dive deeper into the idea of integration and differentiation.
Did you know?
In Latin, the word “Calculus” means pebble.
What is Differentiation and Integration? (H2)
The two major concepts that Calculus is based on are Differential Calculus & Integral Calculus.
Differentiation
Differentiation helps us calculate the rate of change of one quantity relative to another. The resulting rate of change is known as the Derivative. When it comes to finding a derivative of a function f(x), the process of Differentiation comes into play. For example, we have variables x and y, where y = f(x).
The derivative of the function is denoted by f’(x) or dy/dx (dy and dx are known as differentials.)
A limit is an element used in differentiation, which is defined by the degree of closeness to any value that is being approached. For example,
limₓ→2 (x² – 4)/(x – 2) , the limit is read as limit (x) tends to 2. To understand it better, let’s solve it.
limₓ→2 (x² – 4)/(x – 2) = f(x)
Step 1: Factor the numerator & simplify the expression
x² – 4 = (x – 2)(x + 2)
(x – 2)(x + 2)/(x – 2) = x + 2 (for x ≠ 2)
Step 2: Apply the limit
limₓ→2 (x + 2) = 4
The Idea of Differentiation
Think of differentiation as a way of understanding how fast something is changing. For example, imagine you’re racing in your Tesla Roadster on a racing track and Elon Musk is checking how fast you can go. Elon’s personal Teslabots calculate your speed at any instant by taking the derivative of your position over time. If you know how far you drove each second, differentiation tells how your speed is changing, accelerating, or decelerating.
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Click HereDifferentiation Formulas
| Function | Derivative |
|---|---|
|
Constant Rule |
d/dx(c)=0 |
|
Power Rule |
d/dx(xn)=nxn−1 |
|
Sum Rule |
d/dx(f(x)+g(x))=f′(x)+g′(x) |
|
Difference Rule |
d/dx(f(x)−g(x))=f′(x)−g′(x) |
|
Product Rule |
d/dx(f(x)g(x))=f′(x)g(x)+f(x)g′(x)
|
|
Quotient Rule |
d/dx(g(x)/f(x))={f′(x)g(x)−f(x)g′(x)}/g(x)² |
|
Chain Rule |
d/dx(f(g(x)))=f′(g(x))g′(x) |
|
Exponential Functions |
d/dx(ex)=ex |
|
|
d/dx(ax)=axln(a) |
|
Logarithmic Functions |
d/dx(ln(x))=x1 |
|
|
d/dx(loga(x))=xln(a)1 |
|
Trigonometric Functions |
d/dx(sin(x))=cos(x) |
|
|
d/dx(cos(x))=−sin(x) |
|
|
d/dx(tan(x))=sec²(x) |
|
|
d/dx(cot(x))=−csc²(x) |
|
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d/dx(sec(x))=sec(x)tan(x) |
|
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d/dx(csc(x))=−csc(x)cot(x) |
Integration
Integration is used to find the accumulation of change, which is termed the integral. It comes into play when you need to find the area under a curve, the volume of a solid, or even to determine the total rainfall collected in a reservoir. In layman’s terms, Integration helps you find f(x) from f’(x), and this process of finding the integral is called Integration.
∫f(x) dx = Integral for a function
∫The integral sign denotes the summation of infinitesimal quantities.
The relationship between Integration and Differentiation is that both are reciprocals of each other.
Integration also means “adding up small pieces to find the total.” It’s how we find out how much there is of something when it’s changing continuously.
The Idea of Integration
Imagine you’re dreaming of becoming a millionaire by saving money. As you put aside money every day, Integration can tell you your total savings after one month. With further calculation, you can see how long it would take you to become a millionaire.
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Click HereIntegration Formulas
| Function | Integral |
|---|---|
|
Constant Rule |
∫k dx=kx+C |
|
Power Rule |
∫xⁿ dx=(n+1)x/(n+1)+C |
|
Reciprocal Rule |
∫1/x dx= ln x + C |
|
Exponential Functions |
∫ex dx=ex+C |
|
|
∫ax dx=ln(a)ax+C |
|
Trigonometric Functions |
∫sin(x) dx=−cos(x)+C |
|
|
∫cos(x) dx=sin(x)+C |
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∫sec2(x) dx=tan(x)+C |
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∫csc2(x) dx=−cot(x)+C |
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∫sec(x)tan(x) dx=sec(x)+C |
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∫csc(x)cot(x) dx=−csc(x)+C |
|
Integration by Parts |
∫udv=uv−∫vdu |
How to do Well in Calculus?
Understand the difference and connection between integration and differentiation well. Let us simplify the idea of integration and differentiation for you. Differentiation is noticing the change in the thickness of paint as you move the roller. Integration is figuring out the amount of paint used for the entire wall. We hope you have a better idea of how integration and differentiation is connected and how it helps.
To do well in calculus, start with building a solid foundation in Pre-calculus, which includes algebra, trigonometry, inverse trigonometry, analytic geometry, etc, as they are fundamental to calculus. Make sure you don’t just memorise the concepts. Learn and grasp the underlying ideas behind them. Also, try to visualise the concepts using graphs and diagrams to better understand.
Most importantly, understand that there is no shortcut to mastering calculus. You have to practice a lot of problems to get a good hang of the topic. Try to avoid using solution manuals too much to develop your skill in the subject. Also, remember to practice by setting a timer for your calculus practice session, as it can help you with time management during exams.
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Frequently Asked Questions (FAQs)
It’s the reverse of integration and helps us to understand how a quantity changes with respect to another quantity.
For example:
The derivative of the change in position with respect to time gives you velocity.
Firstly, they are inverse operations.
Differentiation breaks down things to understand change.
Integration adds things up to understand totals.
Integrate a function and then differentiate it, you’d get back the original plus a constant.
Understand and visualise the concepts with graphs and diagrams. Remember, calculus is a skill. So, practice regularly. Also, build a strong foundation in Algebra, Trigonometry, and Geometry so you don’t get stuck anywhere. Consider joining the best tuition centre in Kerala, like Xylem Learning, if you’re preparing for exams like IIT-JEE, KEAM, and CUET.
The core motivation for creating calculus was Newton’s objective of solving the problem of planetary orbits. He started by trying to describe the speed of a falling object and found that the speed increases with time. Newton then realised that there was no mathematical concept that could explain this and was driven to create calculus, one of the most influential mathematical tools ever invented.
Calculus was discovered by both Isaac Newton and Gottfried Leibniz independently of each other.