Let
be a function. Suppose that the limit

exists. The limit is denoted by
and is called the **derivative** of
at
.
In this case way say
is **differentiable** at
.

Notice that to say the limit

exists is the same as to say the limit

exists. In this case the two limits are equal.

If
then
Hence

Since
it follows that

Since

we have

for
Therefore

for
Since
we have

It follows from the identity

that

Therefore

It follows from the identity

that

Therefore

Let
be the derivative of the function
then the derivative of the function
is called the **second derivative** of the function
and is denoted by
A derivative of the second derivative is called the **third
derivative** of the function
It is denoted by
Similarly we can define the fourth order
fifth order
and so forth. A derivative of the **nth order** is symbolized by
and is sometimes also written

The above formulae show that the function
has derivatives of all orders and that

This formula is sometimes written

Exercise: Obtain the general formula for the nth order derivative of the
function

Let
Then

In general we see that the nth order derivative takes the form

Exercise: Find the general formula for the nth derivative of function

Let
be a polynomial function of the form

Then

From

we have

Taking derivatives, we have

therefore

Taking derivatives again, we have

therefore

Continuing this way we see that the coefficients
can be obtained from the formula

In the same fashion, we see that if the polynomial
were expressed in the form

then the coefficients
can be obtained from the formula

Exercise: Apply the Expand command of Maple to express these functions as linear combinations of functions of the form

(a)

(b)

(c)

(d)

Exercise: Find the nth derivative of each of the above functions.

Exercise: Find the nth order derivative of the function

These are the nth derivative of the product
of two functions

In general, we have

Suppose that and exist Then

(i)

(ii)

(iii) ( for scalars

If in addition that for all then

(iv)

(v)

(vi) Let be a polynomial function. Then

Exercise: prove each of the above statements.

If
is differentiable at
and
is differentiable at
then the composition
is differentiable at
and

Proof. Define a function
as follows.

We claim that
is continuous at

Let
be given. Since
is differentiable at
there exists
such that

Since
is differentiable at
it follows that
is continuous at
so there exists
such that

Now fix any
with
We
show that

Case 1):
Then

Write
Then
and
and so

therefore

Case 2): Then so clearly

We have proved that

Now
for
we have

even
if
(both sides are zero!) Therefore