Implicit Differentiation
The functions that we have met so far can be described by expressing
one variable explicitly in terms of another variable for example,
Some functions, however, are defined implicitly by a relation between
and such as
1
2
In some cases it is possible to solve such an equation for y as an
explicit function (or several functions) of x. For instance, if we solve
Equation 1 for x, we get
so two of the functions
determined by the implicit Equation 1 are:
and
It’s not easy to solve Equation 2 for explicitly as a function of x by hand.
(A computer algebra system has no trouble, but the expressions it obtains
are very complicated.) Nonetheless, (2) is the equation of a curve called
the folium of Descartes and it implicitly defines as several functions of x.
When we say that is a function defined implicitly by Equation 2, we
mean that the equation
is true for all values of x in the domain of .
Fortunately, we don’t need to solve an equation for in terms of in
order to find the derivative of . Instead we can use the method of
implicit differentiation. This consists of differentiating both sides of the
equation with respect to and then solving the resulting equation for
Example:
Example:
Solution:
Applications
of
Differentiation
1- Mathematical Model for Population Growth or Decay
One model for the growth of a population is based on the assumption that the
population grows at a rate proportional to the size of the population. That is a
reasonable assumption for a population of bacteria or animals under ideal
conditions.
Let’s identify and name the variables in this model:
t = time (the independent variable)
P = the number of individuals in the population (the dependent variable)
The rate of growth of the population is the derivative
So our assumption that the rate of growth of the population is proportional to
the population size is written as the equation
where k is the proportionality constant.
1
2
Example:
A population of protozoa develops with a constant relative growth rate of
0.7944 per member per day. On day zero the population consists of two
members. Find the population size after six days.
Solution:
The relative growth rate is
So,
Then:
Thus:
Example:
A bacteria culture starts with 500 bacteria and grows at a rate proportional
to its size. After 3 hours there are 8000 bacteria.
(a) Find an expression for the number of bacteria after hours.
(b) Find the number of bacteria after 4 hours.
(c) Find the rate of growth after 4 hours.
(d) When will the population reach 30,000?
Solution:
So,
Therefore,
Physicists express the rate of decay in terms of half-life, the
time required for half of any given quantity to decay.
2- Mathematical Model for Newton’s Law of Cooling
Example:
Solution:
Example:
Solution:
3- Mathematical Model for Continuously Compounded Interest
Suppose that $1000 is invested at 6% interest, compounded annually.
After 1 year the investment is worth $1000(1.06) = $1060.
After 2 years it’s worth [$1000(1.06)]1.06 = $1123.60.
After 3 years it’s worth ([$1000(1.06)]1.06)1.06 = $1190.44.
t
After t years it’s worth $1000(1.06) .
In general, if an amount A0 is invested at an interest rate r, then after t
t
years it’s worth A0 (1 r ) .
Usually, however, interest is compounded more frequently, say, n times a
year. Then in each compounding period the interest rate is r n and there
are nt compounding periods in t years, so the value of the investment is:
Example:
Solution:
Example:
Solution:
4- Mathematical Model for Related Rates Problems
In a related rates problem the idea is to compute the rate of change
of one quantity in terms of the rate of change of another quantity
(which may be more easily measured). The procedure is to find an
equation that relates the two quantities and then use the Chain Rule
to differentiate both sides with respect to time.
The given information:
and the unknown:
• In order to express these quantities mathematically, we introduce
some suggestive notation:
Strategy for Solving the Related Rates Problems
1.
2.
3.
4.
5.
6.
7.
Read the problem carefully.
Draw a diagram if possible.
Introduce notation. Assign symbols to all quantities that are
functions of time.
Express the given information and the required rate in terms
of derivatives.
Write an equation that relates the various quantities of the
problem. If necessary, use the geometry of the situation to
eliminate one of the variables by substitution.
Use the Chain Rule to differentiate both sides of the equation
with respect to t .
Substitute the given information into the resulting equation
and solve for the unknown rate.
Example:
Solution:
Example:
Solution:
Example:
Solution:
Example:
Solution:
Example:
Solution:
Example:
Solution:
Example:
A street light is mounted at the top of a 20-ft-tall pole. A man 6 ft tall walks
away from the pole with a speed of 5 ft/s along a straight path. How fast is
the tip of his shadow moving when he is 40 ft from the pole?
Example:
A ladder 5 ft long rests against a vertical wall. If the bottom of the ladder
slides away from the wall at a rate of 1 ft/s, how fast is the angle between the
ladder and the ground changing?
5- The Linear Approximation of a Function
We have seen that a curve lies very close to its tangent line near the
point of tangency. In fact, by zooming in toward a point on the graph
of a differentiable function, we noticed that the graph looks more and
more like its tangent line. This observation is the basis for a method
of finding approximate values of functions.
1
The following linear function whose graph is this tangent line is called
the linearization of f at a.
2
Differentials