Antonio Farfán Vallespín
IMP2007 Introductory Math Course
5.3. Time derivatives and growth rates
When a variable y is a function of time, y=f(t), its instantaneous rate of growth
is defined as:
ry 
dy dt f (t )

y
f (t )
but this last expression is equivalent to:
f (t ) d (ln f (t ))

f (t )
dt
So we can calculate the rate of growth by taking logarithms and differentiating
with respect to time.
This method can be very convenient if f(t) is multiplicative, a quotient or of
exponential type.
Example:
rt
Find the growth rate of V  Ae .
taking natural logarithms
ln V  ln A  rt ln e  ln A  rt
and differentiating with respect to time we find the growth rate:
d (ln V ) d (ln A  rt )

r
dt
dt
Rate of growth of a combination of functions:
Assume the combination of functions y=uv,
u  f (t )
where 
 v  g (t )
taking natural log of y:
ln y  ln u  ln v
and differentiating with respect to time in order to determine the growth rate of
y:
d (ln y ) d (ln u ) d (ln v )


dt
dt
dt
1/2
Antonio Farfán Vallespín
IMP2007 Introductory Math Course
r(uv )  ru  rv
Therefore, the instantaneous rate of growth of the product of two functions is the
sum of their respective rates of growth.
By a similar procedure we can find that the quotient of functions, for instance
y  u v , where u=f(t) and v=g(t), the rate growth will be:
r(uv )  ru  rv
Notation
Sometimes the derivative of a variable y with respect to time is represented with
a dot above the symbol of the variable:
y .
y 
y
t
and as we saw before:
ry 
 ln y  ln y y 1


  y
t
y t y
Therefore, we can express the growth rate of a variable as:
y
ry 
y
BIBLIOGRAPHY
Alpha C. Chiang (1984) Fundamental Methods of Mathematical Economics –Third edition.
McGraw-Hill, Inc. Ch.10
2/2