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Section 2.8The Derivative as a Function
Math 1a
February 13, 2008
Announcements
I Office Hours TW 2–4 in SC 323
I ALEKS is due Wednesday 2/20
I HW on website
Outline
Cleanup: Derivatives of some root functions
The derivative functionWorksheet #1
How can a function fail to be differentiable?
Other notations
The second derivativeWorksheet #2
Last time: Worksheet problems 3 and 4
ProblemLet f (x) = x1/3. Find f ′(x) and its domain.
Answer
f ′(x) =1
3x−2/3. The domain is all numbers except 0.
ProblemLet f (x) = x2/3. Find f ′(x) and its domain.
Answer
f ′(x) =2
3x−1/3. The domain is all numbers except 0.
Last time: Worksheet problems 3 and 4
ProblemLet f (x) = x1/3. Find f ′(x) and its domain.
Answer
f ′(x) =1
3x−2/3. The domain is all numbers except 0.
ProblemLet f (x) = x2/3. Find f ′(x) and its domain.
Answer
f ′(x) =2
3x−1/3. The domain is all numbers except 0.
Last time: Worksheet problems 3 and 4
ProblemLet f (x) = x1/3. Find f ′(x) and its domain.
Answer
f ′(x) =1
3x−2/3. The domain is all numbers except 0.
ProblemLet f (x) = x2/3. Find f ′(x) and its domain.
Answer
f ′(x) =2
3x−1/3. The domain is all numbers except 0.
Outline
Cleanup: Derivatives of some root functions
The derivative functionWorksheet #1
How can a function fail to be differentiable?
Other notations
The second derivativeWorksheet #2
The derivative function
I We have snuck this in: If f is a function, we can compute thederivative f ′(x) at each point x where f is differentiable, andcome up with another function, the derivative function.
I What can we say about this function f ′?
Worksheet #1
Outline
Cleanup: Derivatives of some root functions
The derivative functionWorksheet #1
How can a function fail to be differentiable?
Other notations
The second derivativeWorksheet #2
Differentiability is super-continuity
TheoremIf f is differentiable at a, then f is continuous at a.
Proof.We have
limx→a
(f (x) − f (a)) = limx→a
f (x) − f (a)
x − a· (x − a)
= limx→a
f (x) − f (a)
x − a· limx→a
(x − a)
= f ′(a) · 0 = 0
Note the proper use of the limit law: if the factors each have alimit at a, the limit of the product is the product of the limits.
Differentiability is super-continuity
TheoremIf f is differentiable at a, then f is continuous at a.
Proof.We have
limx→a
(f (x) − f (a)) = limx→a
f (x) − f (a)
x − a· (x − a)
= limx→a
f (x) − f (a)
x − a· limx→a
(x − a)
= f ′(a) · 0 = 0
Note the proper use of the limit law: if the factors each have alimit at a, the limit of the product is the product of the limits.
Differentiability is super-continuity
TheoremIf f is differentiable at a, then f is continuous at a.
Proof.We have
limx→a
(f (x) − f (a)) = limx→a
f (x) − f (a)
x − a· (x − a)
= limx→a
f (x) − f (a)
x − a· limx→a
(x − a)
= f ′(a) · 0 = 0
Note the proper use of the limit law: if the factors each have alimit at a, the limit of the product is the product of the limits.
How can a function fail to be differentiable?Kinks
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Kinks
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Cusps
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Cusps
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Vertical Tangents
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Vertical Tangents
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Weird, Wild, Stuff
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Weird, Wild, Stuff
x
f (x)
x
f ′(x)
Outline
Cleanup: Derivatives of some root functions
The derivative functionWorksheet #1
How can a function fail to be differentiable?
Other notations
The second derivativeWorksheet #2
Notation
I Newtonian notation
f ′(x) y ′(x) y ′
I Leibnizian notation
dy
dx
d
dxf (x)
df
dx
Meet the Mathematician: Isaac Newton
I English, 1643–1727
I Professor at Cambridge(England)
I Philosophiae NaturalisPrincipia Mathematicapublished 1687
Meet the Mathematician: Gottfried Leibniz
I German, 1646–1716
I Eminent philosopher aswell as mathematician
I Contemporarily disgracedby the calculus prioritydispute
Outline
Cleanup: Derivatives of some root functions
The derivative functionWorksheet #1
How can a function fail to be differentiable?
Other notations
The second derivativeWorksheet #2
The second derivative
If f is a function, so is f ′, and we can seek its derivative.
f ′′ = (f ′)′
It measures the rate of change of the rate of change!
Leibnizian notation:
d2y
dx2
d2
dx2f (x)
d2f
dx2
The second derivative
If f is a function, so is f ′, and we can seek its derivative.
f ′′ = (f ′)′
It measures the rate of change of the rate of change!Leibnizian notation:
d2y
dx2
d2
dx2f (x)
d2f
dx2
Worksheet #2