DERIVATIVES
Lecture 4
DERIVATIVES
Lecture 4
NOTATIONS - the ones you need to know how to read aloud
The definition
The
slope
of a line
y =
m
x +b
y = mx
A
B
tan
A
B
x
y
m
0
0
y = mx + b
m is the slope
m = 0
no slope ‘m = ∞’
tan
m
p
o
s
i
t
i
v
e
n
e
g
a
t
i
v
e
a function f
What is the solpe of the secant passing through x ?
h
h
x
f
x
f
m
)
(
)
(
)
h
x
(
f
0
)
x
(
f
0
h
x
0
0
x
h
)
x
(
f
)
h
x
(
f
x
)
h
x
(
y
y
x
y
tan
0
0
0
0
0
)
x
(
dx
df
)
x
(
'
f
h
)
x
(
f
)
h
x
(
f
lim
h
0
0
0
0
0
The derivative of f at point x
0
.
)
x
(
'
f
h
)
x
(
f
)
h
x
(
f
lim
h
0
)
x
(
'
f
x
x
)
x
(
f
)
x
(
f
lim
x
x
0
0
0
0
)
x
(
'
f
x
)
x
(
f
)
x
x
(
f
lim
x
0
0
0
0
DIFFERENT NOTATIONS
0
0
0
x
x
dx
df
)
x
(
f
dx
d
)
x
(
dx
df
h
)
x
(
f
)
h
x
(
f
lim
)
x
(
'
f
h
0
h
)
x
(
f
)
h
x
(
f
lim
)
x
(
'
f
h
0
LEFT-HAND AND RIGHT HAND DERIVATIVES
Wikipedia
f ’(x) is a function which tells us the slope of the function at x.
Tangent Line
The equation of the tangent line to the graph of
f at (c, f(c)) has slope f’(c) and equations.
)
c
x
)(
c
(
'
f
)
c
(
f
y
:
line
Tangent
)
c
)
c
(
'
f
)
c
(
f
(
x
)
x
(
'
f
y
c
)
c
(
'
f
)
c
(
f
b
),
x
(
'
f
m
,
b
x
m
y
When a derivative does not exist...
Differentiabiltity Implies Continuity
If f has a derivative at x = c, then f is continuous x = c
Examples of derivatives
h
x
h
x
h
x
h
)
x
(
f
)
h
x
(
f
0
2
0
2
0
0
0
2
2
x
x
f
0
0
0
0
0
0
2
2
x
h
x
lim
h
)
x
(
f
)
h
x
(
f
lim
h
h
Find the
derivative of
at point x
0
.
Example - how to find the derivative using the definition:
Find the derivative of at x
0
= –3, 0, 5.
2
0
2
0
2
0
0
3
0
3
0
0
0
3
3
3
x
h
h
x
x
lim
h
x
h
x
lim
x
'
f
h
h
3
x
x
f
27
3
3
3
2
'
f
0
0
3
0
2
'
f
75
5
3
5
2
'
f
Example - how to find the derivative using the definition:
(C)’ = 0
The derivative of a constant is zero
c
y
0
y
x
y
1
'
y
x
y sin
x
y cos
'
x
y cos
x
y
sin
'
x
a
y
a
a
y
x
ln
'
x
e
y
x
e
y
'
For all
R
x
BASIC DERIVATIVES
n
x
y
1
'
n
nx
y
,
1
'
1
,
2
1
)'
(
2
x
x
x
x
The derivative of a
x
and the number „e
”
.
1
1
lim
0
h
e
h
h
e
a
a
a
x
x
log
1
)'
(
The number ‘e’ is the real number such that the
slope of the tangent line to the graph of the
exponential function y = e
x
at x = 0 is 1.
(e
x
)’ = e
x
If we take a = e
then
Rule
Chain
)
x
(
'
g
)
x
(
g
'
f
)
x
(
g
(
f
.
Rule
Quotient
g
f
'
g
g
'
f
g
f
.
Rule
Product
'
g
f
g
'
f
g
f
.
'
g
'
f
g
f
.
'
f
c
cf
.
'
'
'
'
'
5
4
3
2
1
2
Basic Properties and formulas
If f (x) and g(x) are differentiable functions (the derivative exists),
c is any real number,
h
x
g
x
f
h
x
g
h
x
f
x
g
x
f
h
0
lim
'
h
x
g
x
f
h
x
g
x
f
h
x
g
x
f
h
x
g
h
x
f
h
0
lim
x
f
h
x
g
h
x
g
h
x
g
h
x
f
h
x
f
h
h
0
0
lim
lim
x
f
x
g
x
g
x
f
'
'
Proof of the Product Rule
QED
g'
f
g
'
f
g
f
'
h
x
g
x
f
h
x
g
h
x
f
x
g
x
f
h
0
lim
'
h
h
x
g
x
g
h
x
g
x
f
x
g
h
x
f
h
0
lim
h
h
x
g
x
g
h
x
g
x
f
x
g
x
f
x
g
x
f
x
g
h
x
f
h
0
lim
h
h
x
g
x
g
x
f
x
g
h
x
g
x
g
x
f
h
x
f
h
]
[
]
[
lim
0
2
)
(
'
'
x
g
x
g
x
f
x
g
x
f
Proof of the Quotient Rule
QED
2
'
g
f
g'
g
'
f
g
f
dx
du
v
dx
dv
u
uv
dx
d
)
(
2
v
dx
dv
u
dx
du
v
v
u
dx
d
OTHER NOTATIONS
For this problem, we must take the
derivative of the
outside
function
, then
multiply by the derivative of the inside
function
.
But the derivative of the inside function is also a chain rule
question.
So, we end up with the following:
f(x) =
sin(cos(x
2
))
f ‘(x) = cos(cos(x))·(- sin(x)) 2x
The Chain Rule - example
Suppose that a function f is continuous and strictly monotone
in the neighbourhood O(a) of point a.
If f ’(a) exists and is never zero, then f
-1
is differentiable at b = f (a)
(a = f
-1
(b)),
)
(
'
1
)
(
'
1
)
(
)'
(
1
1
a
f
b
f
f
b
f
du
dx
dx
du
1
or
DERIVATIVE OF INVERSE FUNCTION
)
b
(
f
x
b
x
df
dx
dx
f
d
1
1
1
DERIVATIVE OF INVERSE FUNCTIONS
0
0
1
0
1
x
'
1
)
y
(
'
1
)
y
(
)'
(
f
f
f
f
The derivative of f
-1
(x) = at point (4,2)
is the reciprocal of the derivative of f (x) = x
2
at (2,4).
x
x
arc
y
cos
x
y
a
log
x
y ln
x
arc
y
sin
x
arc
y
tg
x
arc
y
ctg
a
x
y
ln
1
'
x
y
1
'
2
1
1
'
x
y
2
1
1
'
x
y
2
1
1
'
x
y
2
1
1
'
x
y
x
a
a
0
1
,
0
0
x
2
2
1
1
y
x
y
x
0
1
1
2
2
y
y
0
BASIC DERIVATIVES CD.
2
2
1
1
sin
1
1
cos
1
'
sin
1
)'
sin
(
x
y
y
y
x
arc
y
y
2
sin
1
cos
we choose
y
y
2
sin
1
cos
2
;
2
y
0
cos
y
Because
out of the possibilities
a
x
a
a
a
x
y
y
a
ln
1
ln
1
1
)'
(log
'
PROOFS
2
2
2
1
1
tg
1
1
cos
'
tg
1
)'
tg
(
x
y
y
y
x
arc
2
2
2
2
2
2
1
1
1
1
cos
sin
sin
sin
'
1
)'
(
x
y
ctg
y
y
y
y
ctgy
arcctgx
2
2
1
1
cos
1
1
sin
1
'
cos
1
)'
cos
(
x
y
y
y
x
arc
PROOFS
Example of a function not strictly monotone, continuous at
x = 0, f (0) = 0
with a nonzero derivative f’(0) = 1 and
f
-1
(x)
does not
have a derivative
at 0, because f
-1
(x) is not continuous
there.
M. Gewert, Z. Skoczylas, ‘Analiza matematyczna 1, Definicje, twierdzenia wzory’
The function is
defined as a
uniform limit
NOTE
NOTE
The derivative does not have to be continuous.
If the derivative of f (x) exists
then this implies that f (x) is
continuous.
Proof
A continuous function for which the
derivative
exists and the derivative is not continuous.
(a)
f(x) is differentiable and
(b)
f’(x) is not continuous
-0.1
-0.05
0.05
0.1
-0.004
-0.002
0.002
0.004
-0.1
-0.05
0.05
0.1
-0.075
-0.05
-0.025
0.025
0.05
0
x
0
0
x
x
1
x sin
f is continuous, but has no derivative
f is continuous, has a derivative, but derivative
is not continuous
The derivative of a continuous function doesn't have to be continuous.
We cannot calculate f'(x
0
) by simply calculating f'(x) and putting x →x
0
.
A continuous nowhere differentiable function.
Plot of
Weierstrass Function
over the interval [−2, 2]. The
function has a fractal behavior: every zoom (red circle) is
similar to the global plot.
Weierstrass Function in the complex plane
THE MOST IMPORTANT DERIVATIVES (once more)
A continuous nowhere differentiable function.
A saw-like function with infinite teeth:
First define a saw-tooth function f(x) to be the distance from x
to the integer closest to x. Here's a plot of f:
SPECIFIC DEFINITION
Now define to be . This has as many ‘teeth’ as
f per unit interval, but their height is times the height of the
teeth of f. Here's a plot of , for example:
Finally, define h(x) to be the sum . For every x this
sum converges by
comparison with a geometric series.
It's already beyond elementary calculus to show that h(x) is
continuous (to advanced calculus students: h(x) is the sum of a
uniformly converging series of continuous functions, hence
continuous).
For the proof that h(x) is not differentiable:
the rough idea is that at every step we add more and more
corners. Here's a plot of h(x): (actually only a partial sum
rather than the infinite sum).
( )
( )
( )
( )
( )
( )
0
2
2
2
4
8
2
4
8
n
n
n
g
x
f x
g x
g x
g x
g x
�
=
=
=
+
+
+
+
�
L
Blancmange Function
It is an example of a fractal, in that it is infinitesimally
fractured, and self-similar. No matter how much you
zoom in on a point on the graph, the graph never
flattens out into an approximate non-vertical line
segment through the point.