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2.2c Notes on The Derivative
When we found
y
at a certain x value, we found the slope at that point.
x
𝑓(𝑎+ℎ)−𝑓(𝑎)
.
ℎ
ℎ→0
Another way to write that is 𝑚 = lim
The limit at x = a when h approaches 0.
What if we want to find the slope at any point on the function?
We have to find the derivative.
Derivative:
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
ℎ→0
ℎ
𝑓′(𝑥) = lim
We follow the same method as finding the instantaneous rate of change at a point, but we leave the x ‘s
in the problem!
Other derivative notation:
𝑦 ′ , 𝑓 ′ (𝑥),
𝑑𝑦 𝑑𝑓 𝑑
, , 𝑓(𝑥)
𝑑𝑥 𝑑𝑥 𝑑𝑥
Example 1:
Use the definition of the derivative to find 𝑓 ′ (𝑥), if 𝑓(𝑥) = 𝑥 2 .
Example 2:
Using your answer from example 1, find the slope of 𝑓(𝑥) = 𝑥 2 at 𝑥 = 1, 𝑥 = 0, and 𝑥 = −4.
Example 3:
Find
dy
of 𝑦 = 7𝑥 − 3.
dx
Example 4:
Find 𝑓′(𝑥) if 𝑓(𝑥) = 2𝑥 2 − 5𝑥 + 6
Homework: 2.2c Worksheet #1-8