SM015 - 9.1 (Notes Solutions)

9.1 Derivative of a Function

Friday, July 10, 2020 3:14 PM

Learning Outcomes:
At the end of the lesson, students should be able to
(a) Find the derivative of a function using the first principle.
(b) Discuss the differentiability of a function at a point.

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Differentiation from the First Principle

For a function , its derivative can be obtained by differentiation
from the first principle, i.e.

E.g.
For ,

Topic 9.1 Page 1

Example 1

Tuesday, June 16, 2020 9:37 PM

Find the derivative of the following functions with respect to x by using
first principle.
Solution:
(a)

(b)

Topic 9.1 Page 2

Differentiability of a Function if

Sunday, June 21, 2020 1:40 PM

 A function is differentiable at

exists, i.e.

Remarks: , then is continuous at .
If is differentiable at

Example 2

Given .

Is differentiable at ?
Solution:

is NOT differentiable at

Topic 9.1 Page 3

Example 3

Tuesday, June 16, 2020 9:37 PM

A function is defined by
.

Determine the value of if is differentiable for all real values.
Solution:

is differentiable for all real values
Since is differentiable at

Topic 9.1 Page 4