3. Chain rule
3. Chain rule
3.1 Composite Functions and the Chain Rule
3.2 Inverse Functions and their derivatives
3.1.1 Composite Functions
3.1.2 Inverse Function
DEFINITION: If g and h are two functions on R, the function formed by
first applying function g(x) (inside function) to any number x and then
applying function h(z) (outside function) to the result g(x) is called the
composition of functions g and h and is written as
f (x) = h(g (x))
or
f (x) = (h ◦ g )(x)
(1)
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(2)
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THEOREM: A function f defined on an interval E in R has a well-defined
inverse function on the interval f (E) if and only if f is monotically
increasing on all of E or monotonically decreasing on all of E.
THEOREM: A C1 function f defined on an interval E in R is one-to-one
and therefore invertible on E if f ’(x)>0 for all x ∈ E or f ’(x)<0 for all x
∈ E.
3.1.2 Derivative of the Inverse Function
THEOREM: Let f be a c1 function defined on the interval I in R. If
f’(x)6=0 for all x ∈ I, then:
(a) f is invertible on I,
(b) its inverse g is a C1 function on the interval f (I), and
(c) for all z in the domain of the inverse function g,
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1
f 0 (g (z))
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f (g (z)) = z
for all x in the domain E1 of f, and
(3)
for all z in the domain E2 of g
(4)
x1 6= x2 ⇐⇒ f (x1 ) 6= f (x2 )
3. Chain rule (Inverse Functions and their Derivatives)
g 0 (z) =
g (f (x)) = x
In order for f to have an inverse g, f cannot assign the same point to two
different points in its domain
3.1.2 Chain Rule
Derivative of the outside function h (evaluated at the inside function g)
times the derivative of the inside function g:
d(h ◦ g )
dh
dg
(x) =
(g (x)) ·
(x)
dx
dz
dx
DEFINITION: For any given function f :E1 ⇒ R, where E, the domain of f
is a subset of R, we say the function g :E2 ⇒ R is an inverse of f if
(6)
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(5)
A function f that satisfies (5) on a set E is said to be one-to-one or
injective on E. If f is invertible on its domain, then its inverse is uniquely
defined ⇒ Write f−1 the inverse function of f.
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4. Exponents and Logarithms: f(x)=ax
4.1 The Number ”e”
DEFINITION: The function f (x)=ex is called the exponential function
(and is frequently written as exp(x)) with the letter e reserved to denote
the following irrational number
1 n∼
(7)
e ≡ lim 1 +
= 2.7182818
n→∞
n
THEOREM: As n → ∞, the sequence (1+1/n)n converges to a limit
denoted by symbol e. Furthermore,
k n
lim 1 +
= ek
n→∞
n
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4. Exponents and Logarithms
4. Exponents and Logarithms (Logarithms)
4.2 Logarithms
DEFINITION: The Logarithm of x is the power to which one must raise a
to yield x. It follows immediately from this definition that
aloga (x) = x
and
loga (ax ) = x
(9)
DEFINITION: The Natural Logarithm of x is the power to which one
must raise e to get x. It follows immediately from this definition that
ln(x) = y
⇐⇒
ey = x
(10)
and that
e
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ln(x)
=x
and
x
ln(e ) = x
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(11)
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The graph of y = log(x) is the reflection of the graph of y =10x across the
diagonal {y=x} because, by definition, logarithm function is the inverse
function f −1 of the exponential function.
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4. Exponents and Logarithms
4.3 Properties of EXP and LOG
4. Exponents and Logarithms
4.3.1 Properties of the Exponential Function
(1)
(2)
(3)
(4)
(5)
ar · as = ar +s
a−r = 1/ar
ar /as = ar −s
(ar )s = ars
a0 =1
4.3.3 Constant Elasticity Demand Functions
q = kp 4.3.2 Properties of the Logarithmic Function
(1)
(2)
(3)
(4)
(5)
with
= (p/q)(dq/dp)
(12)
In logarithmic coordinates, demand is now a linear function whose slope is
the elasticity :
ln q = ln kp = ln k + ln p
(13)
log (r · s) = log r + log s
log (1/s) = - log s
log (r /s) = log r - log s
log (r s ) = s log r
log 1 = 0
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4. Exponents and Logarithms
4. Exponents and Logarithms (Derivatives of EXP and LOG)
4.5 Applications
4.4 Derivatives of EXP and LOG
Present Value (PV) to bring all money figures back to the present
THEOREM: The functions ex and ln x are continuous functions on their
domains and have continuous derivatives of every order. Their first
derivatives are given by
(e x )0 = e x
and
(ln x)0 = 1/x
(14)
If u(x) is a differentiable function, then
e u(x)
0
(ln u(x))0 =
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= e u(x) · u 0 (x)
(15)
u 0 (x)
u(x)
(16)
if u(x)>0
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4. Exponents and Logarithms
4.6 Logarithmic Derivative
Derivative of some complex function by using reverse of equation
(16):
u 0 (x) = (ln u(x))0 · u(x)
(17)
Log-Log specification → Slope of the graph of f in log-log
coordinates is the (point) elasticity of f and economists sometimes
write the elasticity as
d(ln f )
(18)
=
d(ln x)
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Continuous Compounding → In an account which compounds
interest continuously at rate r, a deposit of A euros now will yield B =
Aert euros t years from now → PV of B euros t years from now is
Be−rt euros
Annual Compounding → In an account which compounds interest
annually at rate r, a deposit of A euros now will yield B = A(1+r)t
euros t years from now → PV of B euros t years from now is B(1+r)−t
euros
Annuities is a sequence of equal payments at regular intervals over a
specified period of time → PV= A/r (in order to generate a perpetual
flow of A euros a year from a savings account which pays interest
annually at rate r, on must deposit A/r euros into the account
initially)
Optimal holding time
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