The document defines the derivative of a function and explains how to find derivatives. It states that the derivative of a function f(x) at a point a is defined as the limit as h approaches 0 of (f(x+h) - f(x))/h, if the limit exists. This provides a geometric interpretation of the derivative as the slope of the tangent line to the function f(x) at the point a. The document also defines left and right derivatives and notes that for a derivative to exist at a point, the left and right derivatives must be the same at that point. Additionally, it mentions that differentiability requires a function to be continuous at a point.

1. Section 2.8 Derivatives
Section 2.9 The derivative as a function
Learning outcomes
After completing this section, you will inshaAllah be able to
1. explain the definition of derivative of a function
2. find derivatives using definition
( )1
2.8 2.9−
• In the last section we saw that the rate of change of f(x)
at a point ‘x’ is given by
0
( ) ( )
lim
h
f a h f a
m
h→
+ −
=
• Therefore such limits become very important.
• The study of important limit of above type leads to the
concept of derivative, which we study here.

2. Derivative of a function
• Another notation:
dy
dx
• ( )f a′ means derivative at the point a
Geometric interpretation of derivative
Checking differentiability graphically
See examples 1, 2, 3 done in class
See example 4 done in class
The derivative of f(x) is defined by
0
( ) ( )
( ) lim
h
f x h f x
f x
h→
+ −
′ = ,
if the limit exists.
( )2
2.8 2.9−
See class explanation
See class explanation

3. Left/Right derivative of a function
Existence of a derivative
• The next example uses all of these ideas.
End of 2.8, 2.9
The left derivative of f(x) is defined by
0
( ) ( )
( ) lim
h
f a h f a
f a
h−−
→
+ −
′ =
( )3
2.8 2.9−
The right derivative of f(x) is defined by
0
( ) ( )
( ) lim
h
f a h f a
f a
h++
→
+ −
′ =
See explanation given
above for geometric
interpretation of
derivative
The derivative ( )f a′ exists if the left
and right derivatives are same.
Also note the fact that
“if a function is
differentiable at a
point then it is
continuous at that
point”
i.e. the function will surely be not
differentiable at a point where it is
discontinuous
See example 5 done in class