What is differentiation? - Future Study Point

What is differentiation?

What is Differentiation?

What is differentiation?

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 FunctionsChapter 9-Differential Equations
Chapter 2-Inverse Trigonometric FunctionsChapter 10-Vector Algebra
Chapter 3-MatricesChapter 11 – Three Dimensional Geometry
Chapter 4-DeterminantsChapter 12-Linear Programming
Chapter 5- Continuity and DifferentiabilityChapter 13-Probability
Chapter 6- Application of DerivationCBSE 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

Class 12 Maths Important questions on Chapter 7 Integral with Solutions for term 2 CBSE Board 2021-22

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

Chapter 1- Number SystemChapter 9-Areas of parallelogram and triangles
Chapter 2-PolynomialChapter 10-Circles
Chapter 3- Coordinate GeometryChapter 11-Construction
Chapter 4- Linear equations in two variablesChapter 12-Heron’s Formula
Chapter 5- Introduction to Euclid’s GeometryChapter 13-Surface Areas and Volumes
Chapter 6-Lines and AnglesChapter 14-Statistics
Chapter 7-TrianglesChapter 15-Probability
Chapter 8- Quadrilateral

NCERT Solutions for class 10 maths

Chapter 1-Real numberChapter 9-Some application of Trigonometry
Chapter 2-PolynomialChapter 10-Circles
Chapter 3-Linear equationsChapter 11- Construction
Chapter 4- Quadratic equationsChapter 12-Area related to circle
Chapter 5-Arithmetic ProgressionChapter 13-Surface areas and Volume
Chapter 6-TriangleChapter 14-Statistics
Chapter 7- Co-ordinate geometryChapter 15-Probability
Chapter 8-Trigonometry

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-SetsChapter 9-Sequences and Series
Chapter 2- Relations and functionsChapter 10- Straight Lines
Chapter 3- TrigonometryChapter 11-Conic Sections
Chapter 4-Principle of mathematical inductionChapter 12-Introduction to three Dimensional Geometry
Chapter 5-Complex numbersChapter 13- Limits and Derivatives
Chapter 6- Linear InequalitiesChapter 14-Mathematical Reasoning
Chapter 7- Permutations and CombinationsChapter 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

 

 

 

 

 

 

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