The Derivative
While we used the idea of finding tangents using the formula
𝑓 𝑎+ℎ −𝑓 𝑎
𝑓′ 𝑎 = lim
from a specific point 𝑎, 𝑓 𝑎 this
ℎ
ℎ→0
can be generalized to find the slope of the tangent from all
points on a function using the more general form,
𝑓′ 𝑥 =
𝑓 𝑥+ℎ −𝑓 𝑥
lim
ℎ
ℎ→0
.
While we can derive the tangent from an individual point, a
more complex method will lead to the general derivative for
all points on the function. The general derivative statement
speeds up the process if multiple tangents are being found.
A function is differentiable at a point if it exists at that point.
There are three ways a function’s derivative does not exist at a
point.
Cusps or Corners
Vertical Tangents
Discontinuities
(includes jumps, VAs and RDs)
y

x





A normal to a relation
at point 𝑃 is the
perpendicular line (⊥)
to the tangent at 𝑃.
This means the slope of
the normal is the
negative reciprocal of
the tangent’s slope.
y

𝑁𝑜𝑟𝑚𝑎𝑙
x




𝑃

𝑇𝑎𝑛𝑔𝑒𝑛𝑡
We can denote the derivative of a function as 𝑓′, 𝑦′ or
𝑑𝑦
.
𝑑𝑥
1. For the function 𝑓 𝑥 = 2𝑥 − 1 find the value of the
derivative 𝑓′ 𝑎 where 𝑎 = 5.
2. Find the general derivative of the function 𝑦 =
𝑥
.
𝑥+1
3. Find the point(s) where the derivative of the function
𝑦 = 5𝑥 2 − 9𝑥 equals 2.
? pg. 130 #1, 4 – 8, 10, 12