Differentiation
MATHEMATICS FOR SCIENCE
IPSE FPMIPA UPI
There are three notations for the derivative of with respect to
They are
) (x f y
) ( '
' f x
y D
xy
x
dx y dy
dx
d ( )
rule multiple
- constant )
( ' )
( )
3
rule function
power )
2
function contant
0 )
1
1
x cf
x dx cf
d
nx dx x
d dx c
d
n n
rule Quotient
) (
) ( ' ) ( )
( ' ) ( )
( ) (
6)
rule Produc t
) ( )
( .
5)
rule differenc e
- sum
)
4
2 x g
x g
x f x
f x g x
g x f dx
d
x g x f x
g x f
x g x dx f
d
x g x
f x
g x
dx f d
dR Q
Q Q
R
Q Q
Q
dQ Q P d
dQ P Q d
dQ dR
-Q)Q (
R -Q
P PQ
R
2 15
15
2 15
1 15
1
15 15
2
x x
dx x d
x x
dx x x d
dx x d
x dx x
x d dx x
d
cot csc
csc )
5
tan sec
) (sec 4)
sec
) (tan )
3
sin )
(cos )
2 cos
) (sin )
1
2
)
601 4
2 ( )
( x x x
F
60 2
) (
and 1
4 2
) (
y z
y f
x x
y x
g
Imagine trying to find the derivative of
Let
then
) )(
( ))
( (
) (
)
( x z f y f g x f g x
F
or
) )(
( or
) ( ' )) (
( ' )
( '
and at
able differenti
is )) (
( )
(
by defined
function composite
the then
, at
able differenti
is and
at able
differenti is
and If
) (
and )
( Let
dx dy dy
dz dx
dz
y D z
D z
D
x g
x g f
x g
f
x x
g f
x g
f
g f
y
f x
g f
x g y
y f
z
x y
x
3 )
7 3
(
3 7
3
) 7
( )
(
Rule, Chain
the Applying
) ( 7
) (
w rite ),
( Assume
. of
terms in
for solved
be c annot it
, 7
Let
2 2
2 2
3 3
3 3 3 3
x dx y
dy
dx x dy dx
y dy
dx x y d
dx y d
dx d
x x
y x
y
x y y
x y
x y
y
) ( '
) ( ' :
have we
then
. variable ent
independ the
of al differenti the
called ,
. variable t
independen the
of al differenti the
called ,
. variabel
t independen the
of function abel
differenti a
be )
( Let
x dx f
dy
dx x
f dy
y dy
x dx
x
x
f
y
1 1
1
cost average
cost marginal
cost total
cost average
minimum the
Find
2
AC Q MC
Q Q Q C
Q C
Q
Q C Q
C Q Q
Q C dQ
d
C(Q)/Q AC
C'(Q) MC
C(Q) TC
C
MC AC
Q
dx y dy
f dy
dx
y f
x
x f
dx dy
x f
y
1 ) 9
and
) ( then
x of
function monotonic
strictly a
is y where
and ),
( Let
1 1
/dP /dQ
/b dP/dQ
b
•dQ /dP
) b
(
)Q
/b (
/b -b P
P b
•Q b
(Q) f
P f(P)
Q
s s
-
1 1
0 where
1
•
each x) for
y one 17,
p.
function, a
of definition the
(Recall
x.
of value unique
a yield y will
of e given valu a
and
y of value unique
a yield x
of e given valu a
in which one
is function monotonic
•A
1
1 1
0 1
0
1
1 1
.
unknown) one
equation, (one
for x Solve
function inverse
an have ill
function w the
y, of value different
a yield always
x will of
value different
a i.e.,
mapping, one
- to - one a
represents f(x)
y function the
If
1(y) f
x
x f dx
dy
f(x) y
-
This property of one to one mapping is unique to the class of functions known as monotonic functions:
Recall the definition of a function, p. 17,
function: one y for each x
monotonic function: one x for each y (inverse function)
if x1 > x2 f(x1) > f(x2) monotonically increasing
Qs = b0 + b1P supply function (where b1 > 0) P = -b0/b1 + (1/b1)Qs inverse supply function
if x1 > x2 f(x1) < f(x2) monotonically decreasing
Qd = a0 - a1P demand function (where a1 > 0)