Download Derivatives

DIFFERENTIAL CALCULUS
DERIVATIVES
(1)
Slope of a Straight line
Slope ?
.
B
f (b)
𝑟𝑖𝑠𝑒
∆y
f (a)
∆𝑦
Slope = 𝑟𝑢𝑛 = ∆ 𝑥
i. e.
.A
∆𝑦
Slope = ∆ 𝑥 =
𝒇 𝒃 − 𝒇(𝒂)
𝒃−𝒂
For Example:
a ∆x b
A = (1, 2)
B = (3, 5)
∆𝑦
Slope = ∆ 𝑥 =
𝒇 𝒃 − 𝒇(𝒂)
𝒃−𝒂
=
𝟓 −𝟐
𝟑 −𝟏
𝟑
=𝟐
2
Derivative
• How to find the slope of a curve
.
(x + h , f (x + h))
f (x + h)
.
.
.
.. .
f (x)
.
(x , f (x))
x
The slope of the secant line =
Simplifying …
The slope of the secant line =
.
x+h
(x + h , f (x + h))
𝑓 𝑥 + ℎ −𝑓 𝑥
𝑥+ℎ−𝑥
(x , f (x))
.
𝑓 𝑥 + ℎ −𝑓 𝑥
ℎ
3
Derivative
The slope of the secant line =
𝑓 𝑥 + ℎ −𝑓 𝑥
ℎ
As h → 0 :
(x + h , f (x + h))
𝑓 𝑥 + ℎ −𝑓 𝑥
ℎ
ℎ →0
The slope of the secant line = 𝑙𝑖𝑚
.
.
∆y
(x , f (x))
∆x=h
∆y
∆y
𝑓 𝑥 + ℎ −𝑓 𝑥
ℎ
ℎ →0
f ′ (x)= 𝑙𝑖𝑚
“f”
is called the Derivative of
Notations of derivative:
𝒅𝒚
𝒅
f ′ (x) , y ′ , 𝒅𝒙 , 𝒅𝒙 𝒇 𝒙
Example:
Find the slope of the graph of : f (x) = 2 x – 3 as x = 2 applying the definition of the slope of a tangent
line.
Example:
Find the slopes of the tangent lines to the graph of: f (x) = 𝒙 𝟐 + 1 at the points (0, 1) and (– 1, 2)
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