Differentiation and Integration: Definition, Formulas, Problems, & Notes

Updated on Jun 24, 2025 13:41 IST

Differentiation and Integration are essential concepts in Calculus that deal with change and motion. Both are used to determine functions in different situations. Differentiation and Integration are used in Physics, Economics, Engineering, and other streams.

Differentiation is used to calculate the rate of change of a function. Also, using the differentiation calculus, we can calculate velocity, marginal cost, and acceleration. Derivatives of a function are calculated using differentiation.

On the other hand, Integration is used to find the area under the curve. The value calculated using the integration is known as the integral.

Differentiation and Integration are important chapters in Class 12 Maths. These topics have good weight gain in board exams. Also, various entrance tests such as JEE Main, MHT CET, WBJEE and others prepare questions based on differentiation and integration.

Students need to have a good grip on the topics. As we know, in Mathematics, problems are lengthy. So, to calculate the problems quickly, formulas should be at your fingertips. Practice makes perfect, so practice is the key to securing good marks in math.

To guide students with their preparation, the experts at Shiksha have prepared NCERT Solutions. Using Class 12 Maths NCERT Solution, students can practice more questions based on differentiation and integration. In this article, we have defined Differentiation and Integration, formulas, and solved examples.

Also Read:

NCERT Class 12 Notes	NCERT Class 12 Chemistry Notes
Class 12 Maths NCERT Notes	NCERT Class 12 Physics Notes

Table of content

What are Differentiation and Integrations?
What is Differentiation?
What is Integrations?
Differentiation and Integration Formulas
Properties of Integration and Differentiation
Illustrated Examples on Differentiation and Integration
FAQs on Differentiation and Integration

What are Differentiation and Integrations?

In Calculus, differentiation and integration are quite important concepts. Integration proves to be useful while calculating the addition of discrete data, which you will find hard to add one by one or by using a single value. On the other hand, differentiation proves to be useful while observing the changes in the value of a particular quantity with respect to the change in the value of another quantity.

For example, you can evaluate how speed changes according to the value of time, and this is a simple application of differentiation. Integration can be used to determine the length of the cable required to connect two points that are miles apart.

Important Topics:

NCERT Class 11 Note	CBSE Class 11 Physics NCERT Notes
NCERT Class 11 Chemistry Notes	NCERT Class 11 Maths Notes

What is Differentiation?

When the value of a function changes continuously as one of its variables keeps changing, it is known as differentiation. Graphically, it can be shown as a tangent’s slope over a particular point of a function.

Mathematically, differentiation can be expressed as per the expression below:

f’(a) = (f (a+h) - f(a))/h where limit h tends to be 0

If f(x) = y, where y is a function of x, then its derivative is expressed as dy/dx. It is also called as y’s derivative as per x.

Also Read: NCERT Solutions Class 12 Chapter 9 Differentiation

What is Integrations?

A method that is used to find the values of indefinite and definite integrals is called integration. If F(x) is referred to as an integration of f(x), it is given by the following expression:

∫ f(x) dx = F(x) + C

f(x) is referred to as an integrand.

dx is the agent that is used for integration.

C is the constant, and x is the variable.

Also Check: NCERT Solutions Class 12 Chapter 7 Integrals

Differentiation and Integration Formulas

Differentiation Formulas	Integration Formulas
$\frac{d}{d x} (e^{x}) = e^{x}$	$\int_{}^{} e^{x} d x = e^{x} + C$
$\frac{d}{d x} (a^{x}) = a^{x} \log a$	$\int_{}^{} a^{x} d x = \frac{a^{x}}{\log a} + C$
$\frac{d}{d x} (x^{n}) = n x^{n - 1}$	$\int_{}^{} x^{n} d x = \frac{x^{n + 1}}{(n + 1)} + C, n \neq - 1$
$\frac{d}{d x} (k) = 0$	$\int_{}^{} k d x = k x + C$
$\frac{d}{d x} (\ln x) = \frac{1}{x}$	$\int_{}^{} \frac{1}{x} d x = \ln x + C$
$\frac{d}{d x} (\log_{a} x) = \frac{1}{x \ln a}$	$\int_{}^{} \log_{a} x d x = a \log_{a} x - \frac{x}{\ln a} + C$

Properties of Integration and Differentiation

Integration and differentiation are processes that are opposite or inverse to each other.
To determine the values of integration and differentiation, we will have to use the limits within which the function is defined.
Any function’s derivative is always unique, whereas the integrals of a function may or may not be unique.

Details about Differentiation and Integration taught in Class XII. The topic of differentiation and integration is taught in Class XII’s chapter of ‘Integrals’. This chapter has a weigh. tage of 12 marks, which means that this particular topic carries a weightage from 4 to 8 marks in the exam.

Illustrated Examples on Differentiation and Integration

1. y = x³ Find dy/dx

Solution:

We have the formula if y = xⁿ, then dy/dx = nx^n-1

Therefore dy/dx = 3x²

2. If y = tan²x, find dy/dx

Solution:

dy/dx = 2 tanx x d/dx (tanx) = 2tanx x sec²x

3. Calculate dy/dx if y = 1 - tan²x/sec²x

Solution:

Now we have tan²x = sec²x - 1

Therefore, dy/dx = 1 - sec²x + 1/sec²x = 2 - sec²x/sec²x

FAQs on Differentiation and Integration

Q: Give some applications of integration.

A: Integration can be used to find the area that falls between two curves, centre of mass, volume, etc.

Q: Give some applications of differentiation.

A: Differentiation can be used to find the normal and tangent of a given curve, to calculate the minimal and maximum values of a given function, or just to evaluate the rate of a change of a specific quantity with respect to others.

Q: What is differential calculus?

A: Differential calculus is a part of calculus in which we determine the rate at which given quantities vary or change.

Q: Which are the main divisions of calculus?

A: Integral and differential calculus are the two most important divisions of calculus.

Q: What is the other name for integration?

A: Integration is also referred to as antiderivative.