• NEET
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • JEE
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • Class 6-10
      • Class 6th
      • Class 7th
      • Class 8th
      • Class 9th
      • Class 10th
    • View All Options
      • Online Courses
      • Offline Courses
      • Distance Learning
      • Hindi Medium Courses
      • International Olympiad
    • NEET
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • JEE (Main+Advanced)
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • JEE Main
      • Class 11th
      • Class 12th
      • Class 12th Plus
  • NEW
    • JEE 2025
    • NEET
      • 2024
      • 2023
      • 2022
    • Class 6-10
    • JEE Main
      • Previous Year Papers
      • Sample Papers
      • Result
      • Analysis
      • Syllabus
      • Exam Date
    • JEE Advanced
      • Previous Year Papers
      • Sample Papers
      • Mock Test
      • Result
      • Analysis
      • Syllabus
      • Exam Date
    • NEET
      • Previous Year Papers
      • Sample Papers
      • Mock Test
      • Result
      • Analysis
      • Syllabus
      • Exam Date
    • NCERT Solutions
      • Class 6
      • Class 7
      • Class 8
      • Class 9
      • Class 10
      • Class 11
      • Class 12
    • CBSE
      • Notes
      • Sample Papers
      • Question Papers
    • Olympiad
      • NSO
      • IMO
      • NMTC
    • TALLENTEX
    • AOSAT
    • ALLEN e-Store
    • ALLEN for Schools
    • About ALLEN
    • Blogs
    • News
    • Careers
    • Request a call back
    • Book home demo
JEE PhysicsJEE Chemistry
Home
JEE Maths
What Is Differentiation In Maths

What is Differentiation in Maths?

Differentiation is a fundamental concept in calculus dealing with the rate at which a quantity changes. It is the process of finding the derivative of a function, which provides information about the slope of the function at any given point.

1.0Derivatives of Functions 

In differentiation, the derivative of any function, say f(x) at a particular point, represents the rate of change of that function with respect to x at that point. 

It can be denoted as f'(x) or dxd​[f(x)]. 

2.0Types of Differentiation

  1. Basic Differentiation: involves applying simple differentiation rules, such as power, sum, product, and quotient rules, to find the derivative of basic functions.
  2. Successive Differentiation: Refers to finding higher-order derivatives (i.e., second derivative, third derivative, etc.). For example, the second derivative of f(x) is denoted as f′′(x), which represents the rate of change of the rate of change.
  3. Inverse Differentiation: The process of finding the original function from its derivative. This involves techniques such as integration (indefinite integration), where the reverse of differentiation is used to recover the function.
  4. Numerical Differentiation: Used when a function is difficult to differentiate analytically. It involves approximating the derivative using numerical methods such as finite differences.

3.0Rules for Differentiation

  1. Constant Rule: The derivative of a constant C is zero. dxd​[C]=0
  2. Power Rule: dxd​[xn]=nxn−1
  3. Constant Multiple Rule: dxd​[Cf(x)]=C.dxd​[f(x)]
  4. Sum and Difference: dxd​[f(x)±g(x)]=dxd​[f(x)]±dxd​[g(x)]
  5. Differentiation of Product or Differentiation by Parts: The integration by parts formula may also be applied in differentiation, especially in the context of product functions.dxd​[u(x)v(x)]=u′(x)v(x)+u(x)v′(x)
  6. Quotient Rule: 

dxd​[v(x)u(x)​]=[v(x)]2v(x)u′(x)−u(x)v′(x)​

  1. Chain Rule: 

dxd​[f(g(x))]=f′(g(x)).g′(x)

4.0Differentiation of Common Functions

  1. Derivative of Trigonometric Functions: 
  • Differentiation of cos(x) and sin(x):

dxd​[cos(x)]=−sin(x)

dxd​[sin(x)]=cosx

  • Differentiation of tan(x) and cot(x): 

dxd​[tan(x)]=sec2(x)

dxd​[cot(x)]=−cosec2(x)

  • Differentiation of sec(x) and cosec(x): 

dxd​[sec(x)]=sec(x)tan(x)

dxd​[cosec(x)]=−cosec(x)cot(x)

  1. Derivative of Inverse Trigonometric Functions: 
  • dxd​[sin−1(x)]=1−x2​1​
  • dxd​[Cos−1(x)]=−1−x2​1​
  • dxd​[tan−1(x)]=1+x21​
  • dxd​[sec−1(x)]=∣x∣x2−1​1​
  • dxd​[cot−1(x)]=−1+x21​
  • dxd​[cosec−1(x)]=−∣x∣x2−1​1​
  1. Differentiation of Hyperbolic Functions: 
  • dxd​[sinh(x)]=cosh(x)
  • dxd​[cosh(x)]=sinh(x)
  • dxd​[tanh(x)]=sinh2(x)
  • dxd​[sech(x)]=−sech(x)tanh(x)
  • dxd​[cosech(x)]=−cosech(x)coth(x)
  • dxd​[coth(x)]=−cosech2(x)
  1. Logarithmic differentiation:
  • dxd​[log(x)]=x1​
  1. Differentiation of vectors:
  • dxd​[rˉ(t)]=rˉ′(t)

5.0Special Techniques in Differentiation

  1. Differentiation by First Principle: 

f′(x)=h→0lim​hf(x+h)−f(x)​

  1. Partial Differentiation: It is the differentiation of more than one variable. It includes taking the derivative of one variable and taking the other as a constant. 

Partial differentiation examples: x2+y2

∂x∂​[f(x,y)]=2x,∂y∂​[f(x,y)]=2y

  1. Parametric Differentiation: It is used when functions are given in the form of other variables, i.e., x = f(t) and y = g(t). Then, dxdy​=dtdx​dtdy​​
  2. Fractional Differentiation: A generalization of the classical concept of derivatives to non-integer orders. It involves derivatives of any real (or complex) order = not necessarily an integer. Dαf(x)

6.0Differentiation Examples

Problem 1: Differentiate f(x) = logxx. 

Solution: Simplifying the function: 

f(x)=xlog(x) 

Applying the product rule: 

u(x) = x and v(x) = log(x) 

dxd​[xlog(x)]=1×log(x)+x1​×x

dxd​[xlog(x)]=log(x)+1


Problem 2: Differentiate the parametric equations x = t2+1 and y = t3−3t.

Solution: 

dtd(x)​=dtd​[t2+1]=2t

dtd(y)​=dtd​[t3−3t]=3t2−3

Using the Parametric Differentiation: 

dxdy​=dtdx​dtdy​​

dxdy​=2t3(t2−1)​


Problem 3:  Find the partial derivative of f(x,y) = x2y+y3 with respect to x and y.

Solution: Partial derivative with respect to x: 

∂x∂f​=∂x∂​[x2y+y3]=2xy

Partial derivative with respect to y

∂y∂f​=∂y∂​[x2y+y3]=x2+3y2

Table of Contents


  • 1.0Derivatives of Functions 
  • 2.0Types of Differentiation
  • 3.0Rules for Differentiation
  • 4.0Differentiation of Common Functions
  • 5.0Special Techniques in Differentiation
  • 6.0Differentiation Examples

Frequently Asked Questions

Differentiation and integration are inverse functions: differentiation measures the rate of change of a function, whereas integration computes the accumulation of a function's values and vice versa.

The chain rule will be applied to differentiate a composite function by multiplying the outer function derivative and the inner function derivative.

Successive differentiation refers to taking higher-order derivatives, like second, third, and so on, of a function.

Fractional differentiation extends the differentiation concept beyond non-integer orders.

Join ALLEN!

(Session 2025 - 26)


Choose class
Choose your goal
Preferred Mode
Choose State
  • About
    • About us
    • Blog
    • News
    • MyExam EduBlogs
    • Privacy policy
    • Public notice
    • Careers
    • Dhoni Inspires NEET Aspirants
    • Dhoni Inspires JEE Aspirants
  • Help & Support
    • Refund policy
    • Transfer policy
    • Terms & Conditions
    • Contact us
  • Popular goals
    • NEET Coaching
    • JEE Coaching
    • 6th to 10th
  • Courses
    • Online Courses
    • Distance Learning
    • Online Test Series
    • International Olympiads Online Course
    • NEET Test Series
    • JEE Test Series
    • JEE Main Test Series
  • Centers
    • Kota
    • Bangalore
    • Indore
    • Delhi
    • More centres
  • Exam information
    • JEE Main
    • JEE Advanced
    • NEET UG
    • CBSE
    • NCERT Solutions
    • Olympiad
    • NEET 2025 Results
    • NEET 2025 Answer Key
    • JEE Advanced 2025 Answer Key
    • JEE Advanced Rank Predictor

ALLEN Career Institute Pvt. Ltd. © All Rights Reserved.

ISO