MCV4U1: Unit#2 – Lesson #1 Name:
2.1 The Derivative Function (First Principles)
Warm-up: Determine the slope of the tangent to 𝑦 = √𝑥 − 3 at 𝑥 = 7 .
Lesson:
A. What does the Derivative Function mean?
1. The slope of the tangent line to the graph of y = f (x) at the general point P(a, f (a)) is given by m = f '(a) .
2.
The instantaneous rate of change (IRC) in the variable y with respect to the variable x, where y = f (x), at x = a is given by: IROC = f '(a).
Notation: 𝒚 ′
(y prime), 𝒇′(𝒙) ( f prime at x), 𝒅𝒚 𝒅𝒙
= 𝒅 𝒅𝒙 𝒇(𝒙) (dy by dx) (We will alternate these notations.)
The process of finding the derivative of a function is called differentiation.
B. How do we find the Derivative Function mean? - First Principles
First Principles is the process of differentiation (finding the derivative function) by computing the limit:
1.
Use First Principles to find the derivative functions of a) ( )
 x

b) y

2
3 x

1
Note: There is a shortcut to doing derivatives that we will see throughout this unit.

MCV4U1: Unit#2 – Lesson #1 Name:
2.
Determine the slope of the line tangent to y

2
3 x

1
at each of the following values of x. a) x = 0 b) x = 1
3.
Complete the following table 𝑓(𝑥) = 𝑥
Determine the derivative with respect to x of each of the following functions:
Write the derivative 𝒇′(𝒙) by itself.
What pattern do
With x as the base (ie 1 =1x 0 ). c) x =2 𝑓(𝑥) = 𝑥
2 𝑓(𝑥) = 𝑥
3 you see?
B. Use the pattern to predict the derivative of 𝑓(𝑥) = 𝑥 777
. C. What do you think would be for where n is a positive integer?
Notes:
1. If a function f is not continuous at x = a then the function f is not differentiable at x = a .
2. If a function f is continuous at x = a then the function f may be or not differentiable at x = a. How do you know? If you cannot draw a unique, defined tangent at “a” then the function is not differentiable at “a”.
3 ways derivatives fail to exist:
If the function is not differentiable, we say that the derivative does not exist .
Homework: Page 73 #1, 4, 7b, 9, 14, 16, 19