WEEK ONE DIFFERENTIATION 1

WEEK ONE

DIFFERENTIATION 1

CONTENTS

(a) Limits of a Function

(b) Differentiation from First Principle

(c) Differentiation of Polynomials

SUB TOPIC: LIMITS OF A FUNCTION

A function is a value that describes how a value of the variable is maniplutated to generate a value of an equation Other symbols that can be used for functions includes Note that all functions are rules but not all rules are functions.

Example:

Let y where can take non – negative (real values). Now let use power of 10 for to find the value of When

From the above, you observe that, as becomes very large, the value of gets very close to 1.

Hence,[mediator_tech]

There are functions that their limit does not exist.

Example:

. The limit of this function does not exist, since division by zero is impossible.

CLASS ACTIVITY:

1. Calculate and . Hence, conclude on the two.
2. If calculate
3. Define a function. (b). justify the statement: “All functions are rules but all rules are not functions”

SUB TOPIC: DIFFERENTIATION FROM FIRST PRINCIPLE

This is the derivative of a function from the point of limiting value of a function. The derivative of is defined as: if the limit exists. Other notation for derivative is

Note that, the change that take place in as produces a corresponing change in

Example:

1. Find from first principle, the derivative of ² where are constants.

Solution:

Let the change in

²

²

²²]

²²

²²]

²²²

²²²

²

Hence, the general derivative of a given function is denoted as

1. Find the derivative of ² using first principle.

Solution:

Let the change in

²]²

²²]²

²²]²

²

Note:

The derivative of any constant is zero.

The derivative of n is ^{n – 1}

In conclusion, the following steps are important in finding derivative using first principle.

1. Find
2. Find
3. Find
4. Find if it exists.

Examples:

1. Find the derivative of each of the following
2. ⁵
3. ³
4. ^1/3

SOLUTIONS:

1. ⁵ ⇒ ^{5 – 1 4}
2. ³ ⇒ ²
3. ^1/3 ⇒ ^2/3

CLASS ACTIVITY:

1. Find the derivative of ^{– 2} using first principle.
2. Find from first principle
3. ² b. ²

Show that the derivative of ⁿ is ^{n – 1}

1. Find the derivative of the following functions:
2. ^-7 b. ^1/2 c. ⁴

SUB TOPIC: DIFFERENTIATION OF POLYNOMIALS

To differentiate a polynomial, we differentiate term by term.

Example:

1. Differentiate ⁴³²

Solution:

⁴³²³²

1. If ⁵⁴³²

Solution:

^{5432 432}

CLASS ACTIVITY:

1. If ⁴³², differentiate term by term
2. If ⁸⁶², differentiate term by term

PRACTICE EXERCISE:

Objectives:

1. Another name of differentiation is (a) gradient (b) function (c) reverse.
2. The derivative of y = 4x + 3 is? (a)12 (b) 7 (c) 4.
3. Evaluate . (a)7 (b)-27 (c) 18.

Essay:

1. Find from first principle, the derivative, with respect to of ²
2. By first principle ²
3. If
4. Define function.
5. Differentiate with respect to x; .

ASSIGNMENT

1. Evaluate .
2. Examine .
3. Find
4. Find from the first principle, the derivative of .
5. If

KEY WORDS

• LIMIT
• CONTINUITY
• FUNCTION
• DERIVATIVE
• DIFFERENTIATION
• FIRST PRINCIPLE
• WITH RESPECT TO
• POLYNOMIALS
• GRADIENT