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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 as 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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If p(x) is a
polynomial of degree n (i.e. the largest exponent in the
polynomial) then,
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We will need to be careful with the “non-prime” notation for
derivatives. Consider each of the
following.
The presence of parenthesis in the exponent denotes
differentiation while the absence of parenthesis denotes exponentiation.
Collectively the second, third, fourth, etc. derivatives are called higher
order derivatives.
Let’s take a look at some examples of higher order
derivatives.
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Example 1 Find
the first four derivatives for each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
(a)
There really isn’t a lot to do here other than do the
derivatives.
Notice that differentiating an exponential function is
very simple. It doesn’t change with
each differentiation.
[Return to Problems]
(b)
Again, let’s just do some derivatives.
Note that cosine (and sine) will repeat every four
derivatives. The other four trig
functions will not exhibit this behavior.
You might want to take a few derivatives to convince yourself of this.
[Return to Problems]
(c)
In the previous two examples we saw some patterns in the
differentiation of exponential functions, cosines and sines. We need to be careful however since they
only work if there is just a t or
an x in argument. This is the point of this example. In this example we will need to use the
chain rule on each derivative.
So, we can see with slightly more complicated arguments
the patterns that we saw for exponential functions, sines and cosines no
longer completely hold.
[Return to Problems]
|
Let’s do a couple more examples to make a couple of points.
|
Example 2 Find
the second derivative for each of the following functions.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
(a)
Here’s the first derivative.
Notice that the second derivative will now require the
product rule.
Notice that each successive derivative will require a
product and/or chain rule and that as noted above this will not end up
returning back to just a secant after four (or another other number for that
matter) derivatives as sine and cosine will.
[Return to Problems]
(b)
Again, let’s start with the first derivative.
As with the first example we will need the product rule
for the second derivative.
[Return to Problems]
(c)
Same thing here.
The second derivative this time will require the quotient
rule.
[Return to Problems]
|
As we saw in this last set of examples we will often need to
use the product or quotient rule for the higher order derivatives, even when
the first derivative didn’t require these rules.
Let’s work one more example that will illustrate how to use
implicit differentiation to find higher order derivatives.
|
Example 3 Find
for
Solution
Okay, we know that in order to get the second derivative
we need the first derivative and in order to get that we’ll need to do
implicit differentiation. Here is the
work for that.
Now, this is the first derivative. We get the second derivative by
differentiating this, which will require implicit differentiation again.
This is fine as far as it goes. However, we would like there to be no
derivatives in the answer. We don’t,
generally, mind having x’s and/or y’s in the answer when doing implicit
differentiation, but we really don’t like derivatives in the answer. We can get rid of the derivative however by
acknowledging that we know what the first derivative is and substituting this
into the second derivative equation.
Doing this gives,
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Now that we’ve found some higher order derivatives we should
probably talk about an interpretation of the second derivative.
If the position of an object is given by s(t) we know that the velocity is the
first derivative of the position.
The acceleration of the object is the first derivative of
the velocity, but since this is the first derivative of the position function
we can also think of the acceleration as the second derivative of the position
function.
Alternate Notation
There is some alternate notation for higher order
derivatives as well. Recall that there
was a fractional notation for the first derivative.
We can extend this to higher order derivatives.