Higher Order Derivatives
Let’s start this section with the following function.
By this point we should be able to differentiate this function without any problems. Doing this
we get,
Now, this is a function and so it can be differentiated. Here is the notation that we’ll use for that,
as well as the derivative.
This is called the
second derivative
and
is now called the
first derivative
.
Again, this is a function so we can differentiate it again. This will be called the
third
derivative
. Here is that derivative as well as the notation for the third derivative.
Continuing, we can differentiate again. This is called, oddly enough, the
fourth derivative
.
We’re also going to be changing notation at this point. We can keep adding on primes, but that
will get cumbersome after awhile.
This process can continue but notice that we will get zero for all derivatives after this point. This
set of derivatives leads us to the following fact about the differentiation of polynomials.
Fact
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View Full DocumentIf
p(x)
is a polynomial of degree
n
(
i.e.
the largest exponent in the polynomial) then,
We will need to be careful with the “nonprime” notation for derivatives. Consider each of the
following.
The presence of parenthesis in the exponent denotes differentiation while the absence of
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 Fall '08
 sc
 Derivative, higher order derivatives

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