Higher Derivatives

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Lesson Priority: High

Calculus $\longrightarrow$
Discovering Derivatives $\longrightarrow$
 
Objectives
  • Learn what compound derivatives mean and how we usually name and notate them
  • See (but not usually use) the limit definition of the second derivative
  • Understand important patterns and relationships between the function and its derivative, second derivative, third derivative, etc.
Lesson Description

Knowing derivatives is one thing, but what about the derivative of the derivative? What does that describe about a function? What about the derivative or the derivative of the derivative? This lesson formally discusses higher derivatives and in which situations we care about their use and interpretation.

 
Practice Problems

Practice problems and worksheet coming soon!

 

Down the Rabbit Hole

In Calculus, when we refer to higher derivatives, we are referring to taking the derivative of an object multiple times. For example, simply stated, the second derivative is the derivative of the derivative. I'm sure as you read that you're wondering - oh boy, what about the derivative of the derivative of the derivative? Or the derivative of the derivative of the derivative of the derivative? Yes, it's possible to do those things. Yes, those things have names. No, we will not commonly go down that seemingly never-ending rabbit hole.For better or worse, beside the first derivative with which we are already familiar, only the second derivative (the derivative of the derivative) has intrinsic meaning and is commonly asked for. There is nothing stopping us from continually taking the derivative of the derivative of the derivative (of the derivative, of the derivative, etc.) until our hearts explode, but such results have little meaningful interpretation relative to the original function you started with.
Vocab FYI
When we need to look at the derivative of the derivative of a function, we call that object the second derivative of the function.

The Second Derivative - Limit Definition

As stated, the second derivative is the derivative of the derivative. Formally, we can define it as a limit of the instantaneous rate of change of the first derivative.
Definition:Let $f'(x)$ be the derivative of function $f(x)$. The second derivative, notated $f''(x)$, is defined as$$\lim_{h \to 0} \frac{f'(x+h) - f'(x)}{h}$$
Hopefully this looks very familiar as an echo of the limit definition of the first derivative - after all, the formula is nearly identical. Because of that similarity, it is very important that you can make the connection between what this formula says, and what the original derivative limit definition formula says. Fortunately, we will not use the limit definition of the second derivative for practical purposes, just like we don't routinely use the limit definition of the first derivative to differentiate expressions.What we should walk away from this limit definition with is a basic understanding of the interpretation of the second derivative. Recall from early exploratory lessons on derivatives that the interpretation of the first derivative is instantaneous rate of change. Since the second derivative is the derivative of the derivative, the second derivative is therefore a measure of how the derivative is changing. Graphically, this will translate to a property called concavity - we will study this soon in more detail soon (see inflection points and concavity »). For now, what we need to know is how to calculate second derivatives when we're asked to find them.

The Second Derivative - Practical Use

Definition:Let $f'(x)$ be the derivative of $f(x)$. Then$$\frac{d}{dx} \,\, f'(x) = f''(x)$$is the second derivative of $f(x)$.
In other words, to find the first derivative, just take the derivative of the derivative, using the derivative rules and shortcuts you know for the particular function in question.
 
Example 1Find $f''(x)$ for$$f(x) = x^4 + 3x^3$$$\blacktriangleright$ The second derivative is the derivative of the derivative. So to find it, first we'll take the derivative, and then we'll take the derivative of that result.$$f'(x) = 4x^3 + 9x^2$$$$f''(x) = 12x^2 + 18x$$
 

Understanding Higher Derivatives

Definition:Let $f^{(n)}(x)$ be the $n$-th derivative of $f(x)$. Then$$\frac{d}{dx} \,\, f^{(n)}(x) = f^{(n+1)}(x)$$is the $n+1$-th derivative of $f(x)$.
Formally, we are simply saying that multiple derivatives means taking the derivative iteratively.
 
Example 2Find the second and third derivatives of $f(x) = 4x^2 + e^x$.$\blacktriangleright$ First, take the derivative:$$f'(x) = \frac{d}{dx} \,\, \big[ 4x^2 + e^x \big] = 8x + e^x$$Now, the second derivative will be the derivative of the derivative.$$\Rightarrow f''(x) = 8 + e^x$$Finally, the third derivative will be the derivative of the second derivative.$$\Rightarrow f'''(x) = e^x$$
You Should Know
Second derivatives have a natural interpretation in Physics - if you are given a function that describes an object's position as a function of time, then the function's first derivative is the object's velocity function, and the second derivative is the object's acceleration function. The third derivative of position with respect to time is not commonly studied, but is interpreted as "jerk". This third derivative is zero if acceleration is assumed constant, which is the case in most high school and early college Physics courses.
We will use higher derivatives throughout our Calculus journey, though, as mentioned, second derivatives are commonly the highest one you'll be asked to work with.

Higher Derivatives - Practical Use

So what is it that you actually need to know? Practically, teachers and professors will ask you at some point to calculate something like the second derivative or the third derivative (or, if they woke up on the wrong side of the bed, the fourth or fifth derivative). Unfortunately, there is no formulaic shortcut. If you are asked for the second derivative, your first task is to understand what that means - the derivative of the derivative. Your second task is to execute the calculation by first calculating the derivative, and then calculating the derivative of that result (thus calculating the derivative of the derivative).
 
Example 3Find the second derivative of the function $f(x) = x^4 - 3x^3 - 7$.
Show solution
$\blacktriangleright$ To execute this problem, find the derivative of the function, then find the derivative of that result.First Derivative:$$\frac{d}{dx} \,\, \big[ x^4 - 3x^3 - 7 \big] = 4x^3 - 9x^2$$Second Derivative:$$\frac{d}{dx} \,\, \big[ 4x^3 - 9x^2 \big] = 12x^2 - 18x$$
 
Example 4Find the second derivative of the function $g(x) = -4x^5 + e^{2x}$.
Show solution
$\blacktriangleright$ Just like the previous problem, we seek to take the derivative twice.First Derivative:$$\frac{d}{dx} \,\, \big[ -4x^5 + e^{2x} \big] = -20x^4 + 2e^{2x}$$Second Derivative:$$\frac{d}{dx} \,\, \big[ -20x^4 + 2e^{2x} \big] = -80x^3 + 4e^{2x}$$
 
Example 5Find the third derivative of the function $h(x) = \frac{2}{x^2} + e^{-x} - \sin(x)$.
Show solution
$\blacktriangleright$ This isn't too different from the last two examples, except that we will be taking the derivative a third time to get the answer we desire.First Derivative:$$\frac{d}{dx} \,\, \big[ 2x^{-2} + e^{-x} - \sin(x) \big]$$$$= -\frac{4}{x^3} - e^{-x} - \cos(x)$$Second Derivative:$$\frac{d}{dx} \,\, \big[ -4x^{-3} - e^{-x} - \cos(x) \big]$$$$=\frac{12}{x^4} + e^{-x} + \sin(x)$$Third Derivative:$$\frac{d}{dx} \,\, \big[ 12x^{-4} + e^{-x} + \sin(x) \big]$$$$=-\frac{48}{x^5} - e^{-x} + \cos(x)$$
If you found that you mixed up any details in these examples, make sure you're practiced up on the basic function derivative functions in the lessons on polynomial derivatives » , exponential derivatives » , and trig function derivatives » .

Higher Derivatives - Notation

In the definitions above, you've already seen some of the notation we use for higher derivatives. But beside knowing how to find high derivatives conceptually, it is also our job to recognize and/or use the proper notation when working with any number of compound derivatives. Recall that for "regular" (aka "first") derivatives, we had two notation systems », both of which we're expected to know: The prime notation and the differential notation. Let's look at both of these again, widening our perspective to allow for higher derivatives.Prime NotationRecall that for a function $f(x)$, we notate its derivative function using the "prime notation" as $f'(x)$. The convention with higher derivatives is to use multiple primes to notate higher derivatives, up to three or four, and after that, to replace the primes with a number in parenthesis. Here's what it looks like.First derivative:$$f'(x)$$
Second derivative:$$f''(x)$$
Third derivative:$$f'''(x)$$
Fourth derivative:$$f''''(x)$$or$$f^{(4)}(x)$$(Some teachers start the parenthesis notation at four, some at five, but either way people will know what you mean.)
Fifth derivative:$$f^{(5)}(x)$$
Sixth derivative:$$f^{(6)}(x)$$
$n$-th derivative:$$f^{(n)}(x)$$
Differential NotationRecall that for a function $f(x)$, we often are asked for its derivative in the differential notation that looks like$$\frac{d}{dx} \,\, f(x)$$Additionally, some teachers insist on a shorthand single fraction, of the form$$\frac{df}{dx}$$Though this is more popular when the function uses $y=$ form:$$\frac{dy}{dx}$$We will notate higher derivatives in differential notation form by applying exponents to the expression. This might seem a little weird, but there is good reason to do this, and specifically, good reason to do it in the particular way that we do it. In the numerator of the differential fraction, apply the exponent to the $d$. In the denominator, apply the exponent to the differentiating variable. Here are some examples. If $y = f(x)$, then the first several derivatives of $y$ will be notated as follows.First derivative:$$\frac{dy}{dx}$$
Second derivative:$$\frac{d^2 y}{dx^2}$$
Third derivative:$$\frac{d^3 y}{dx^3}$$
Fourth derivative:$$\frac{d^4 y}{dx^4}$$
And so on.
Pro Tip
It is our job to understand each notation, as we will definitely see each along our way, and we are absolutely expected to recognize and interpret each notation. As you gain experience in Calculus, you'll better understand which of these notations is more convenient in which situations.

Higher Derivatives for Polynomials

One thing you may have noticed is that when you take the derivative of a polynomial of degree $n$, the result is a new polynomial of degree $n-1$. This is a direct implication of the Power Rule », since it literally involves decreasing polynomial terms' exponents by $1$. Since the degree of a polynomial is determined by the highest power in the polynomial, and since its derivative will result in a polynomial with each term one power lower than its corresponding original term, we have the following result.
Definition:Let $f(x)$ be a polynomial of degree $n$. Then$$\frac{d}{dx} \,\, f(x) = f'(x)$$will be a polynomial of degree $n-1$. Furthermore,$$\frac{d^{n+1}}{dx^{n+1}} \,\, f(x) = 0$$
In words, if you take one more derivative than the degree of the polynomial, your answer is a guaranteed $0$.
Pro Tip
As we learn more complicated derivative techniques such as the upcoming Product Rule » and Chain Rule », taking second (or occasionally third) derivatives will be a certain but infrequent task, and the task is not always a one-step simple solution. For now, focus on understanding that being asked for higher derivatives simply means we're being asked to take the derivative of the derivative multiple times. You'll have plenty of chance to practice higher derivatives of more complicated objects in each of those forthcoming lessons.

Put It To The Test

 
Example 6For the function $g(x)$, notate its fourth derivative in two different ways.
Show solution
$\blacktriangleright$ The clearest two choices here are to show the derivative using the prime notation and the differential notation.Prime Notation:$$g''''(x) \,\,\, \mathrm{or} \,\,\, g^{(4)}(x)$$Differential Notation:$$\frac{d^4}{dx^4} \,\, g(x)$$
 
Example 7Find the second derivative of $f(x) = 2x^3 + \frac{x^6}{2}$.
Show solution
$\blacktriangleright$ Remember that "second derivative" means take the derivative twice. We will find the derivative of the expression, and then take the derivative of that result.$$\frac{d}{dx} \,\, \left[ 2x^3 + \frac{x^6}{2} \right] = 6x^2 + 3x^5$$$$\frac{d}{dx} \,\, \left[ 6x^2 + 3x^5 \right] = 12x + 15x^4$$$$\therefore \frac{d^2}{dx^2} \,\, f(x) = 12x + 15x^4$$
 
Example 8Find the second derivative of $f(x)=e^x - 3x^8$.
Show solution
$\blacktriangleright$ We will proceed the same way as we did in the last problem.$$\frac{d}{dx} \,\, \big[ e^x - 3x^8 \big] = e^x - 24x^7$$$$\frac{d^2}{dx^2} \,\, \big[ e^x - 3x^8 \big] = \frac{d}{dx} \,\, \big[ e^x - 24x^7 \big]$$$$=e^x - 168x^6$$
 
Example 9Find the second derivative of $f(x)=5^x - e^{x/2}$.
Show solution
$$\blacktriangleright \,\, \frac{d}{dx} \,\, f(x) = 5^x \cdot \ln(5) - \frac{1}{2} \, e^{x/2}$$$$\frac{d^2}{dx^2} \,\, f(x) = 5^x \cdot \left( \ln(5) \right)^2 - \frac{1}{4} \, e^{x/2}$$
 
Example 10Find the third derivative of $y= \sin(x) + \ln(x)$.
Show solution
$\blacktriangleright$ The only thing different this time is that we will take the derivative of the original expression three times.$$\frac{dy}{dx} = \cos(x) + x^{-1}$$$$\frac{d^2 y}{dx^2} = -\sin(x) -x^{-2}$$$$\frac{d^3 y}{dx^3} = -\cos(x) + 2x^{-3}$$$$=-\cos(x) + \frac{2}{x^3}$$
 
Lesson Takeaways
  • Understand what it means when we refer to higher derivatives
  • Specifically, be ready and able to find the second derivative of any function or expression by taking the derivative of the derivative
  • Although unlikely, be prepared to take the third, fourth, or higher derivative of a function or expression by knowing how to do so iteratively
  • Fluently recognize and be able to write higher derivatives using proper notation in either prime form or differential form
  • Recognize the fact that each derivative of a polynomial function yields a polynomial that is one degree smaller, and that the $n+1$-th derivative of an $n$ degree polynomial is consequently zero

Lesson Metrics

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Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

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Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

You Should Know - Somewhat elective information that may give you a broader understanding.

Warning! - Something you should be careful about.

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