The differential calculus was introduced sometime during 1665 or 1666, when Isaac Newton first concieved the process we now know as differentiation (a mathematical process and it yields a result called derivative). Among the discoveries of Newton and Leibnitz are rules for finding derivatives of sums, products and quotients of composite functions together with many other results. In this lesson we define derivative of a function, give its geometrical and physical interpretations, discuss various laws of derivatives and introduce notion of second order derivative of a function.

OBJECTIVES

After seen this video, you will be able to :

• define and interpret geometrically the derivative of a function y = f(x) at x = a;
• prove that the derivative of a constant function f(x) = c, is zero;l
• find the derivative of f n  from first principle and apply to find the derivativesÎ =(x) x ,ⁿn E Q  of various functions;

• state and apply the results concerning derivatives of the product and quotient of two functions;

•   state and apply the chain rule for the derivative of a function;

•   find the derivative of algebraic functions (including rational functions); and find second order derivative of a function.

EXPECTED BACKGROUND KNOWLEDGE

• Binomial Theorem

•  Functions and their graphsn

•   Notion of limit of a function.

DERIVATIVE OF A FUNCTION

VELOCITY AS LIMIT

CHECK YOUR PROGRESS 26.1

GEOMETRICAL INTERPRETATION OF dy/dx

Let y = f (x) be a continuous function of x, draw its graph and denote it by APQB.

DERIVATIVE OF CONSTANT FUNCTION

 Statement : The derivative of a constant is zero.

Proof : Let y = c be a constant function. Then y = c can be written as

DERIVATIVE OF A FUNCTION FROM FIRST PRINCIPLE

Recalling the definition of derivative of a function at a point, we have the following working rule for finding the derivative of a function from first principle:

CHECK YOUR PROGRESS 26.2

ALGEBRA OF DERIVATIVES

Many functions arise as combinations of other functions. The combination could be sum, difference, product or quotient of functions. We also come across situations where a given function can be expressed as a function of a function.

 In order to make derivative as an effective tool in such cases, we need to establish rules for finding derivatives of sum, difference, product, quotient and function of a function. These, in turn, will enable one to find derivatives of polynomials and algebraic (including rational) functions.

DERIVATIVES OF SUM AND DIFFERENCE OF FUNCTIONS

CHECK YOUR PROGRESS 26.3

DERIVATIVE OF PRODUCT OF FUNCTIONS

You are all familiar with the four fundamental operations of Arithmetic : addition, subtraction, multiplication and division. Having dealt with the sum and the difference rules, we now consider the derivative of product of two functions.

CHECK YOUR PROGRESS

QUOTIENT RULE

You have learnt sum Rule, Difference Rule and Product Rule to find derivative of a function expressed respectively as either the sum or difference or product of two functions. Let us now take a step further and learn the “Quotient Rule for finding derivative of a function which is the quotient of two functions.

CHECK YOUR PROGRESS

CHECK YOUR PROGRESS