Example: f (x) = x2 is a continuous, continuously
differentiable function .
Example: f (x) = 1 /x is not continuous at x = 0 . (Furthermore, its
derivative is also not
continuous at that point.)
Example: f (x) = l x l is a continuous function that is not continuously
differentiable: its
derivative is 
when x is negative ,
when x is positive, and undefined at x = 0 .
Because it’s
still fairly well behaved, you might say that it is piecewise continuously
differentiable if you
really wanted to.
In economics, the term marginal means the effect of a small change in one
thing on
something else, like the marginal utility of consumption or the
marginal
product of labor. Looking back, we see that this fits the definition of a
derivative
quite well. You'll probably become familiar with these:
Utility function: U =U(c)
Marginal utility of consumption: dU / dc or U' (c)
Production function: Y = F(L)
Marginal product of labor: dF / dL or p · F' (L)
The derivative can also be inter preted as the slope of a line tangent to the
function at
that point. Think back to diagrams of total cost and marginal cost curves.
Calculating the derivative of a function using the proper definition can be very
tedious. The quick and easy way is to recall the power rule :
Because you can often break down functions (like polynomials) into several terms
of
this form, you can take most derivatives easily using this. Here are some other
rules
to fol low for taking derivatives:
| Rule |
Functional form |
Derivative |
| Addition rule : |
 |
| Product rule: |
| Quotient rule: |
| Chain rule: |
| Inverse rule: |
(The quotient rule and the inverse rule require that the
term in the denominator is
not zero, obviously.) The function 
in the
chain rule is called a composite
function; that is, is not directly a
function of x, but it is a function f of g(x) . If
that sounds confusing, think of this example: your utility is not actually a
function of
prices. However, prices do affect how much you can afford to buy, and that
affects
your utility. In economics, this is called an in dividual ’s value function :
V(p) =U(x(p))
In other words, the “value” of facing prices p is the utility you get from your
optimal
demand x(p) when facing these prices.
The inverse rule is also very useful for getting information from inverse
functions.
For example, a consequence of utility maximization is an equation like “the
marginal
utility of con suming some good equals the marginal cost (that is, price) of that
good”:
U' (x) = λ p
( λ is some constant, the multiplier from the utility maximization
problem—ignore it
for now.) This gives us an inverse demand function for x very easily:
(Ignore the constant λ .) We might want to know how demand changes when prices
change—in other words, what is the derivative of the demand function, dx/ dp ?
This
inverse demand function tells the opposite derivative:
We can use the inverse rule to determine what we want:
Let’s actually take this one step further , and try to
establish the price elasticity of
demand. For small changes in prices, this is defined as
We have a formula for dx / dp , so we can stick this into
the equation. We also have
the condition that p = λU' (x) , so we’ll substitute that in as well . The
final answer is:
This formula always works (more or less). If we know x and
we know the utility
function, we can always calculate the price elasticity this way.
As a final note, Some interesting functions have x as an exponent or take
the
logarithm of x. ( Two functions that show up frequently are exponential
e and the
natural logarithm 
.) Remember these
rules for exponents and logarithms:
The power rule makes it easy to take the derivatives of
most functions. However,
these “interesting” functions—like sine, cosine, and logarithm—have derivatives
that
aren’t so simple. For natural logarithms and exponentials , here are the rules:
If you encounter a trigonometric function or something
stranger and need its
derivative, consult Chapter 3 of Sydsæter, Strøm, and Berck (or another math
book)
for a list.
