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

$f'\left ( x \right )=\lim_{h\rightarrow 0}\frac{f\left ( x+h \right )-f\left ( x \right )}{h}$

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

$f'\left ( e^{x} \right )=e^{x}$

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

$f\left ( x \right )=\frac{u\left ( x \right )}{v\left ( x \right )}$

$f'\left ( x \right )=\frac{u'\left ( x \right )v\left ( x \right )-u\left ( x \right )v'\left ( x \right )}{\left ( v\left ( x \right ) \right )^{2}}$

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

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 System	Chapter 9-Areas of parallelogram and triangles
Chapter 2-Polynomial	Chapter 10-Circles
Chapter 3- Coordinate Geometry	Chapter 11-Construction
Chapter 4- Linear equations in two variables	Chapter 12-Heron’s Formula
Chapter 5- Introduction to Euclid’s Geometry	Chapter 13-Surface Areas and Volumes
Chapter 6-Lines and Angles	Chapter 14-Statistics
Chapter 7-Triangles	Chapter 15-Probability
Chapter 8- Quadrilateral

NCERT Solutions for class 10 maths

Chapter 1-Real number	Chapter 9-Some application of Trigonometry
Chapter 2-Polynomial	Chapter 10-Circles
Chapter 3-Linear equations	Chapter 11- Construction
Chapter 4- Quadratic equations	Chapter 12-Area related to circle
Chapter 5-Arithmetic Progression	Chapter 13-Surface areas and Volume
Chapter 6-Triangle	Chapter 14-Statistics
Chapter 7- Co-ordinate geometry	Chapter 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-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

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What is differentiation?