3.6 Calculating Higher-Order Derivatives

Welcome back to AP Calculus with Fiveable! You’re on the last key topic of unit 3, pat yourself on the back. 🥳

✏️ Higher-Order Derivatives

The term “higher-order derivative” may seem intimidating at first, don’t you think?

Fear not! This simply means that we can take the first derivative, second derivative, third derivative, and so on of a function. This is useful because graphically and analytically, first and second derivatives can provide us with all sorts of helpful information. 〽️

🔄 The first derivative of a function tells us where f(x) has a relative minimum or maximum, because the slope, f’(x), is equal to zero.

🔀 The second derivative of a function provides us with information about the concavity of a function by allowing us to find points of inflection, the location where f(x) changes from concave up to concave down and vice versa.

We can denote the second-order derivative using any of the following notation below:

Similarly, we can denote higher-order derivatives using the following notation:

Now that we know what they are, how exactly do we calculate higher-order derivatives (we’ll call them HODs from this point onwards for convenience)?

🧮 Calculating Higher-Order Derivatives: A Step-by-Step Guide

🌶️ Key Takeaway: Calculating second, third, fourth, and nth derivatives down the road follows the same process as taking the first derivative of f(x)! With the second derivative, however, we’ll take the derivative of f’(x). With the third derivative, we’ll take the derivative of f’’(x). Notice the pattern?

Let’s work through an example! Say, we’re looking at a function f(x) below:

Using the Power Rule, our handy dandy best friend from Unit 2 when we started learning differentiation, we find the first derivative—f’(x)—of our function f(x):

As mentioned earlier, the second derivative of f(x) is the same thing as the derivative of f’(x)! That means we can use the Power Rule on f’(x)…

And if we keep going: third derivative = derivative of f’’(x) = second derivative of f’(x), whichever makes more sense to understand:

Since f’’’(x) is a constant, the fourth derivative onwards of f(x) will be 0. In a classroom setting, you will typically be asked to find the first, second, or even third derivatives of a given function.

Now that we’ve walked through the general process of finding HODs, let’s go through some practice questions! 😌

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💭 Higher-Order Derivatives: Let’s Practice!

Prompt: For each of the following functions f(x), find the second derivative f’’(x)!

1️⃣ Set 1: Quick Power Rules and Trig

Example 1:****

First, solving for first derivative via Power Rule where 6 * 4 = 24, -2 * 2 = -4, and 5 * 1 = 5:

Rinsing and repeating the Power Rule, this time for f’(x), for the second derivative where 24 * 3 = 72 and -4 * 1 = -4:

…and we’re done! Example 2:****

Trigonometric functions are among the handful of functions in which you just have to know their respective derivatives. When in doubt, feel free to check out our study guide on Trig Function Derivatives as a refresher. 💫

In our case, sin(x) and cos(x) happen to alternate with each other! Whenever you change from cosine to sine, don’t forget to change the sign, so if we’re deriving cos(x), we’ll get -sin(x), and if we’re deriving -cos(x), we’ll get sin(x).

On the other hand, we’ll keep the sign the same whenever we derive sin(x)—which is the case for our first derivative:

As briefly mentioned already, deriving cos(x) means that we’ll have to change signs. Right now, we have a positive term, which then makes the derivative term a negative one! 👌

Example 3:****

In this example, we’re still working with trigonometric functions, but we also have a composite function here! As a refresher, composite functions are functions inside of other functions. This is where we see “inner” and “outer” functions (sometimes notated as f(g(x)), but we’ll write as O(I(x)) to avoid confusing the Fs).

Here, our outer function (O(x)) is cos(x), and our inner function (I(x)) is 2x. When working with composite functions, we’ll have to use the Chain Rule: ⛓️

To set things up:

Plugging this above:

We’ll repeat this process to find the second derivative. This time, our O(x) = -2sin(x) and I(x) = 2x.

Again:

2️⃣ Set 2: Chain and Product Rule Galore

Need more practice with the Chain Rule? Worry not—here’s some more problems to work with! 🧠 Example 4:****

Our outer function, O(x), will be x²; our inner function, I(x), will be the four terms inside the parenthesis.

Using Power Rule for O(x) and I(x):

Then plugging back into the Chain Rule recipe:

This sounds like a lot of work, don’t you think? Instead of multiplying out every single term, we can use the Product Rule, treating the two parenthesis terms (we’ll call them L(x) and R(x) to represent “left” and “right” terms, respectively) separately, to make our lives easier!

Putting them all together:

Quite honestly, you can keep the terms as is, but if you want to expand them for fun and practice, you’ll get the following answer:

Phew! 🔥 Example 5:****

Don’t let the square root over the whole expression fool you! We can rewrite f(x) as:

Now that f(x) looks more familiar and approachable, we can do the usual Chain Rule shenanigans:

Let’s now use Product Rule! To derive the left term, we’ll repeat the Chain Rule process above. The right term is more straightforward as we’ll only do classic ol’ Power Rule. Try this out on your own, and you should get the following:

Tying it all together:

Again, feel free to leave your answer in unsimplified form; what matters most when working through these examples is that you’re aware of which rules to apply depending on the type of function we’re working with! ⭐

3️⃣ Set 3: Rational Functions and Natural Logs

Example 6:****

The derivative of tan(x) is sec²(x), while the derivative of ln(x) is 1/x. Adjusting due to Chain Rule for our first term gives us the following first derivative:

This time, we can write the equation as follows to make the problem more Chain Rule-friendly in our brains:

The derivative of sec(x) is sec(x)tan(x). Let’s apply Chain and/or Power Rule to the rest of the terms to get our final answer:

Example 7:****

Since we’re working with the quotient of two functions, we can use the Quotient Rule:

Use the Quotient Rule again (and the Chain Rule when deriving the denominator):

And that’s a wrap - hopefully, this helps you feel more confident calculating higher-order derivatives! While you may find that some of the examples took more work to solve, higher-order derivative problems provide much more practice in differentiation as a whole. 🥂