What is differentiation?
Differentiation is the derivative of a function with respect to an independent variable, a derivative of the function is a change in a function with respect to an independent variable.In class 9 you have studied the polynomial like y=x² +2,y = x +1, y = 2x² + x + 2 etc, where x is the independent variable and y is the function of x means the value of y depends on x therefore we can rewrite these polynomial equations in the form as an example f(x) = x² +2. If we draw a graph between f(x) and x , and consider an infinitesimal change in x is Δx then a change in function is Δf(x), therefore differentiation of f(x) with respect to x is determined by Δf(x)/Δx, So differentiation of a function is the ratio between the change in the function to a change in an independent variable which is written as dy/dx or f ‘(x) in calculus.
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Continuity and Differentiability of a function
What is differentiation?
Table of content
- What is differentiation?
- Derivatives of a function as limits
- Formula of differentiation
- Rules of differentiation
- Application of differentiation
What is differentiation?
Differentiation is the derivative of a function with respect to an independent variable. Differentiation in mathematics is the measurement of per unit change in the function with respect to an independent variable.
Let there is a function y = f(x)
The per-unit change in the function with respect to an independent variable ‘x’ is given by dy/dx is called differentiation of y with respect to x.
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Derivatives of a function as limits
Let there be a real values function f(x) and x is a point on its defined domain,let ‘h’ is an infinitesimal change (i.e nearest to 0) then the derivative of the function, f ‘(x) is given by
Formula of differentiation
f ‘(sin x) = cos x
f ‘(cos x) =-sin x
f ‘(tan x) = sec²x
f ‘(cot x) = -cosec²x
f ‘(sec x) = tan x sec x
f ‘(cosec x) = -cot x cosecx
f ‘(log x) = 1/x
Rules of differentiation
Sum or difference rule
Let f(x) = u(x) ± v(x)
Then f ‘( x) = u'(x) ± v'(x)
Product rule
Let f(x) = u(x)× v(x)
Then f ‘( x) = u'(x)v(x) + v'(x)u(x)
Quotient rule
Application of differentiation in real life
With the help of differentiation, we can determine the rate of change of a quantity with respect to another qantity, if one quantity is the function of another quantity.
- Velocity is the rate of change of the displacement, if displacement is S and time is t then velocity(v) is given by v=ds/dt.
- Acceleration is the rate of change of the velocity, if velocity is v and time is t then acceleration(a) is given by a=dv/dt.
- Differentiation helps in determining the maxima and minima of a curve.
- It also helps in determining the tangent of the curve.
NCERT Solutions of Science and Maths for Class 9,10,11 and 12
NCERT solutions for class 12 maths
Chapter 1-Relations and Functions | Chapter 9-Differential Equations |
Chapter 2-Inverse Trigonometric Functions | Chapter 10-Vector Algebra |
Chapter 3-Matrices | Chapter 11 – Three Dimensional Geometry |
Chapter 4-Determinants | Chapter 12-Linear Programming |
Chapter 5- Continuity and Differentiability | Chapter 13-Probability |
Chapter 6- Application of Derivation | CBSE Class 12- Question paper of maths 2021 with solutions |
Chapter 7- Integrals | |
Chapter 8-Application of Integrals |
Class 12 Solutions of Maths Latest Sample Paper Published by CBSE for 2021-22 Term 2
Class 12 Maths Important Questions-Application of Integrals
Solutions of Class 12 Maths Question Paper of Preboard -2 Exam Term-2 CBSE Board 2021-22
Solutions of class 12 maths question paper 2021 preboard exam CBSE Solution
NCERT Solutions for class 9 maths
NCERT Solutions for class 10 maths
CBSE Class 10-Question paper of maths 2021 with solutions
CBSE Class 10-Half yearly question paper of maths 2020 with solutions
CBSE Class 10 -Question paper of maths 2020 with solutions
CBSE Class 10-Question paper of maths 2019 with solutions
NCERT Solutions for class 11 maths
Chapter 1-Sets | Chapter 9-Sequences and Series |
Chapter 2- Relations and functions | Chapter 10- Straight Lines |
Chapter 3- Trigonometry | Chapter 11-Conic Sections |
Chapter 4-Principle of mathematical induction | Chapter 12-Introduction to three Dimensional Geometry |
Chapter 5-Complex numbers | Chapter 13- Limits and Derivatives |
Chapter 6- Linear Inequalities | Chapter 14-Mathematical Reasoning |
Chapter 7- Permutations and Combinations | Chapter 15- Statistics |
Chapter 8- Binomial Theorem | Chapter 16- Probability |
CBSE Class 11-Question paper of maths 2015
CBSE Class 11 – Second unit test of maths 2021 with solutions