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# 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

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

### 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

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

### 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

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

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

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 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