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Roberto’s Notes on Differential Calculus
Chapter 4 : Basic differentiation rules Section 7
Higher order derivatives
What you need to know already: What you can learn here:
Basic differentiation rules. How to repeat the process of differentiation to
obtain derivatives of derivatives, that is,
higher derivatives.
Once we have computed the derivative of a function y = f ( ) x , we end up
with another function y = f ( ) x. Why not take the derivative of this new function,
and then its derivative and so on?
Because we have better things to do with our time!
But what if that repetition provides something useful? And it does, so let us set
the notation and terminology for this simple concept.
Definition
The derivative of the derivative of a function f ( x )is
called the second derivative of that function and is
denoted by one of the symbols:
f ( ) x y
2
2
d y
dx
Why is the exponent of 2 placed differently on the numerator and denominator?
There is a reason for that, but it is linked to certain operations that are done in
advanced calculus. For now, think of it just as a strange quirk with which you need
to live! And being ambitious, we do not stop here.
Definition
The derivative of the second derivative of a function
f ( x ) is called the third derivative of that function
and is denoted by one of the symbols:
f ( ) x y
3
3
d y
dx
Now the game can be continued, so as to define the fourth, fifth, …, n - th
derivative of a function, for any integer number n.
Definition
Repeating the process of differentiation n times
generates the n-th derivative of a function, which is
denoted by one of the symbols:
( )
n
f x
( n )
y
n
n
d y
dx
If n 2 , then the n - th derivative is also said to be a
higher derivative of order n.
Now you have changed the notation again: numbers in brackets?
I’d love to take credit for this notation, but it is one that has been used
universally for a long time. After the third derivative, we would need to insert many
prime symbols and they would be difficult to count. Adopting the notation of
exponents in brackets keeps things easier.
OK, but what is all this stuff useful for, besides being a perversely fun game?
It turns out that these higher derivatives have many concrete meanings in
applications. Remember that a derivative indicates a rate of change, so each higher
derivative represents the rate of change of the previous one. The most common and
famous applications of this idea are for graphs and motion functions.
Technical fact
The second derivative of a function y = f (^) ( x )
represents the rate at which the slope changes. This
is called the concavity and we shall analyze it later in
more detail.
Technical fact
If x = x t ( )is a function of time that describes the
position of a moving object, then:
1. The first derivative x t ( ) represents the
velocity of the object.
2. The second derivative x ( ) t represents the
acceleration of the object.
3. The third derivative x^ ( ) t represents the jerk
of the object.
Proof
Just think of the meaning: the second derivative tells us how fast the velocity
changes and that is acceleration. The third derivative tells us how fast the
acceleration changes and that tells us how jerky the motion is.
How to use even higher derivatives will become apparent in the study of
infinite series.
I can wait! For now, can we see how the game is played?
Example: ( )
2
x
f x
x
To compute the first derivative of this function, we begin with the quotient
rule, followed by addition, power and constant rules for the numerator, and
linear for the denominator:
Example: ( )
f x
x
In order to compute the derivatives of this function, it is convenient to write it
as a negative power, so as to avoid the quotient rule.
( ) ( )
1 f x 3 x 1
− = −
Now let us compute its higher derivatives and keep the coefficients separate,
as we did before:
( ) ( )
2 f x 1 3 x 1 3
− (^) = − −
( ) ( ) ( )( )
(^3 ) f x 1 2 3 x 1 3
− (^) = − − −
( ) ( )( )( )( )
(^4 ) f x 1 2 3 3 x 1 3
− = − − − −
( ) ( ) ( )( )( )( )( )
(^4 ) f x 1 2 3 4 3 x 1 3
− = − − − − −
Notice that we have a pattern developing, since there are:
As many negatives as the order of the derivative
A factorial equal to the order of the derivative
An exponent given by the negative of the order minus 1
A power of 3 equal to the order of the derivative.
This pattern will continue, so that:
( ) ( ) ( ) ( )( )
1 1! 3 1 3
n n^ n n f x n x
− − = − −
Notice that if we let n = 0 and assume 0!=1, this formula becomes:
( ) ( ) ( ) ( )( ) ( )
0 0 0 1^01 f x 1 0! 3 x 1 3 3 x 1
− − − = − − = −
So the “0-th” derivative is the original function, as it should be!
You will see more examples of this method in the future. For now, it is time to
practice on the basics you have seen here.
Summary
Higher derivatives are obtained by successively computing the derivative of a lower order derivative.
The order of a derivative refers to how many times differentiation has been performed, starting from the original function.
For simple functions, higher order derivatives may develop a pattern that can be summarized in a single formula, often including a factorial number.
Common errors to avoid
When looking for a pattern for the higher derivatives of a function, don’t stop too soon: you may need at least 5-6 derivatives before it becomes clear.
Learning questions for Section D 4- 7
Review questions:
- Describe what a higher derivative is.
- Describe the notation used for higher derivatives.
Memory questions:
- Present two correct notations for the second derivative of a function y = f (^) ( x )
- Present two correct notations for the n - th derivative of a function y = f (^) ( x )
- What is the physical meaning of the second derivative of a position function?
- What information does the second derivative contain about the graph of a
function?
- Which formula represents n! as a product?
- What is the value of 0!?
Computation questions:
In questions 1- 22 , compute the second and, if not too mind-boggling, third derivative of the given function.
2 y = 12 x + 7 x
y 12 x
y x
x y x
3 2 3 x x 5 y x
2 3
2
3 x x x y x
y
x
x y
x
x y x
- What is the relation between factorials and higher derivatives?
- What quadratic expression represents the value of
( )
( )
n
n
?
Proof questions:
- Assume that f (^) ( x )is a function with both first and second derivative. What is the second derivative of the function ( )
2 y = x f x? Collect like terms and factors in the
final answer.
- When computing the general formula for
( )
n
f x , you may run into a product of the form 3 5 7 ( 2 n − 1 ), that is, the product of the first n odd numbers. For our
purposes, it is enough to denote such product as I have just done, but you may wonder if there is a factorial-based way to represent this number. Of course it is NOT ( 2 n −1 !),
since such formula includes all even numbers, and some authors denote it by ( 2 n −1 !!) ( double factorial ). But there is a nice formula for it:
1
2 1! 1 3 5 7 2 1 2 1!
n
n n n
−
− − = −
For some weekend fun, you task is to show that this formula is correct by using the method of induction, that is, by showing that:
a) The formula works for n =1.
b) If the formula works for any n , it also works for n + 1.
Application questions:
- What is the acceleration of an object moving on the x axis so that its position at
time t is
3 x^ =^2 t −^5?
- An object falls through a force field so that its height is given by the function
( )
( )
2
h t
t
in meters and seconds. Compute the formula that provides
the acceleration of this object in terms of time.
What questions do you have for your instructor?
