# Differentiation

### Have you ever inflated a spherical balloon?

What do we observe? As we start inflating the balloon its radius starts increasing and consequently, its volume also starts increasing.

Volume *V* of the spherical balloon is given by, V=.

If r=1 cm then V = .

If r=2 cm then V= .

If we want to find the rate at which the volume V increases as the radius r increases, then we want to find the rate of change of Volume with respect to the radius.

So, if we want to find the **rate of change of a dependent variable** (Volume V here) **with respect to independent variable** (radius r here), we want to find or (r).

Here we will introduce the concept of **Differentiation.**

### I. Derivative of a function at a point.

Let f(x) be a real valued function and x = c be any point in its domain, then the derivative of f(x) at x= c is denoted by (c) and it is defined by f(c)=lim provided this limit exists.

### II. Derivative of a function in general.

In general, the derivative of f(x) is denoted by (x) and it is defined by

(x)=lim provided this limit exists.

Also, if *y* = *f*(*x*), then the **derivative or the differential coefficient of f(x)** with respect to

*x*is denoted by or or (x).

The process of finding the derivative of a function is called Differentiation.

**To differentiate a function (x)**** means to find its derivative(x)****.**

### Derivative of some standard functions

### Algebra of Derivative of functions

If ** f(x)** and

**be any two functions such that their derivatives are defined over the common domain then**

*g*(*x*)- [f(x)+g(x)]=f(x)+g(x)
- [f(x)-g(x)]=f(x) – g(x)
- [f(x)g(x)]=g(x)f(x) + f(x)g(x)

This is also known as **Product Rule of Differentiation.**

- = where g(x) 0.

This is also known as **Quotient Rule of Differentiation**

### **Examples**

**Examples**

Now let’s consider some **examples** on differentiation .

** Example 1:** Find the derivative of f(x)= + 3x.

(x)=() +3(x)=11+3(1)=11+3

** Example 2: **Find the derivative of f(x)=7Sin x-3Tan x.

(x)=7(Sin x) -3(Tan x)=7Cos x – 3 x

** Example 3:** Find the derivative of f(x)=Sin xCos x .

(x)=(Sin xCos x)

Cos x(Sin x) + Sin x(Cos x)= Cos x(Cos x) + Sin x(-Sin x)**using Product rule.**

= x – x

** Example 4:** Find the derivative of f(x)= .

(x)=

== using Quotient Rule.

==

** Example 5:** Find the derivative of f(x)=+20+15.

(x)=()+20()+(15)=

= +20(12)+0=+240.

## Check Point

Find the derivative of the following functions *f*(*x*) with respect to *x*.

- f(x)=+20x
- f(x)=Cosx -3Tan x
- f(x)=Sinx Tanx
- f(x)=
- f(x)=5 + 2 + 27

#### Answer Key

- 15 + 20
- -Sinx – 3 x
- Cosx(Tan x)+Sin x( x)
- 5 + 34

## Personalized Online Tutoring

eTutorWorld offers affordable one-on-one live tutoring over the web for Grades K-12, Test Prep help for Standardized tests like SCAT, CogAT, MAP, SSAT, SAT, ACT, ISEE and AP. You may schedule online tutoring lessons at your personal scheduled times, all with a Money-Back Guarantee. The first one-on-one online tutoring lesson is always FREE, no purchase obligation, no credit card required.

For answers/solutions to any question or to learn concepts, take a **FREE TRIAL **Session.

*No credit card required, no obligation to purchase. *

*Just schedule a FREE Sessions to meet a tutor and get help on any topic you want!*

## Pricing for Online Tutoring

Tutoring Package | Validity | Grade (1-12), College |
---|---|---|

5 sessions | 1 Month | $124 |

1 session | 1 Month | $25 |

10 sessions | 3 months | $239 |

15 sessions | 3 months | $354 |

20 sessions | 4 months | $449 |

50 sessions | 6 months | $1049 |

100 sessions | 12 months | $2049 |

## IN THE NEWS

Our mission is to provide high quality online tutoring services, using state of the art Internet technology, to school students worldwide.

Tutoring

Test Prep

Worksheets

Pricing

About Us

Blog

Free Trial

Login

Terms of service

Privacy Policy

Money Back Guarantee

Technical requirements

FAQs

Job Opportunities

Sitemap

**Connect with us**

+1-269-763-4602

+1-269-763-5024

**Online test prep and practice**

SCAT

CogAT

SSAT

ISEE

PSAT

SAT

ACT

AP Exam

**Science Tutoring**

Physics Tutoring

Chemistry Tutoring

Biology Tutoring

**Math Tutoring**

Pre-Algebra Tutoring

Algebra Tutoring

Pre Calculus Tutoring

Calculus Tutoring

Geometry Tutoring

Trigonometry Tutoring

Statistics Tutoring

**Quick links**

Free Worksheets

Fact sheet

Sales Partner Opportunities

Parents

Passive Fundraising

Virtual Fundraising

Our Expert Tutors

Safe and Secure Tutoring

Interactive Online Tutoring

After School Tutoring

Elementary School Tutoring

Middle School Tutoring

High School Tutoring

Home Work Help

Math Tutors New York City

Press

©2022 eTutorWorld Terms of use Privacy Policy Site by Little Red Bird

©2022 eTutorWorld

Terms of use

Privacy Policy

Site by Little Red Bird