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2.2d Notes on Differentiability
One-Sided Derivatives:
Just as we have one-sided limits, there are also one-sided derivatives.
The definition of one-sided derivatives is the same as two-sided derivatives:
lim
h0
f ( x  h)  f ( x )
h
and
lim
h0
f ( x  h)  f ( x )
h
A function is differentiable at x = a iff:
lim 𝑓(𝑥) = lim+𝑓(𝑥) = 𝑓(𝑎) and lim−
𝑥→𝑎 −
𝑥→𝑎
𝑥→𝑎
𝑓(𝑎+ℎ)−𝑓(𝑎)
ℎ
= lim+
𝑥→𝑎
𝑓(𝑎+ℎ)−𝑓(𝑎)
ℎ
In other words, the function needs to be continuous at a and the derivatives on both sides must be
equal.
Example 1:
Determine if f(x) is differentiable at x=0.
 x2 , x  0
f ( x)  
 2 x, x  0
Is f (x) continuous at 0?
One-sided derivatives:
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Example 2:
Find the derivative of f(x) = |x| at x = 0.
−𝑥, 𝑥 ≤ 0
𝑓(𝑥) = {
𝑥, 𝑥 > 0
There are four instances where the derivative will not exist at a point.
Vertical Tangent
Corner
Cusp
Discontinuity
Example 3:
Where is the function graphed to the right not continuous?
Where is the function graphed to the right not differentiable?
Homework: 2.2d Worksheet #1-15
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